Lecture 4.
Continuity of functions
1. Continuity of a function at a point
Definition 1.Let the function y=f(x) is defined at the point x 0 and in some neighborhood of this point. Function y=f(x) is called continuous at point x 0 , if there is a limit of the function at this point and it is equal to the value of the function at this point, i.e.
Thus, the continuity condition for the function y=f(x) at the point x 0 is that:
Because 
, then equality (32) can be written in the form
(33)
This means that for finding the limit of a continuous functionf(x) you can go to the limit under the function sign, i.e. into function f(x) instead of an argument x substitute its limit x 0 .
lim sin x\u003d sin (lim x);
lim arctg x\u003d arctg (lim x); (34)
lim lоg x\u003d log (lim x).
The task.Find the limit: 1) 
;
2)
.
Let us give a definition of the continuity of a function, based on the concepts of an argument increment and a function.
Because conditions 
and 
are the same (Fig. 4), then equality (32) takes the form:
or 
.
Definition 2. Function y=f(x) is called continuous at point x 0 , if it is defined at the point x 0 and its surroundings, and the infinitesimal increment of the argument corresponds to the infinitesimal increment of the function.
The task. Investigate the continuity of a function y=2x 2 1.
Properties of functions that are continuous at a point
1. If the functions f(x) and φ
(x) are continuous at the point x 0, then their sum 
, composition 
and private 
(given that 
) are functions continuous at the point x 0 .
2. If the function at=f(x) is continuous at the point x 0 and f(x 0)\u003e 0, then there exists a neighborhood of the point x 0 in which f(x)>0.
3. If the function at=f(u) is continuous at the point u 0, and the function u \u003d φ (x) is continuous at the point u 0 \u003d φ (x 0 ), then complex function y=f[φ (x)] is continuous at the point x 0 .
2. Continuity of the function in the interval and on the segment
Function y=f(x) is called continuous in the interval (a; b) if it is continuous at each point of this interval.
Function y=f(x) is called continuous on the segment
[a;
b] if it is continuous in the interval ( a;
b), and at the point x=and is continuous on the right (i.e. 
), and at the point x=b is continuous on the left (i.e. 
).
3. Breakpoints of a function and their classification
The points at which the continuity of the function is broken are called break points this function.
If a x=x 0 function discontinuity point y=f(x), then at least one of the conditions of the first definition of the continuity of a function is not satisfied in it.
Example.
1.
. 2.
3)
4)
.
▼ Break point x 0 is called a discontinuity point first kind function y=f(x) if at this point there exist finite limits of the function on the left and on the right (one-sided limits), i.e. 
and 
... Wherein:
The value | A 1 -A 2 | are called jump function at the break point of the first kind. ▲
▼ Break point x 0 is called a discontinuity point second kind function y=f(x) if at least one of the one-sided limits (left or right) does not exist or is equal to infinity. ▲
The task. Find breakpoints and find out their type for functions:
1)
;
2)
.
4. Basic theorems on continuous functions
The theorems on the continuity of functions follow directly from the corresponding theorems on the limits.
Theorem 1. The sum, product and quotient of two continuous functions is a continuous function (for the quotient, except for those values \u200b\u200bof the argument in which the divisor is not zero).
Theorem 2. Let the functions u=φ (x) is continuous at the point x 0, and the function y=f(u) is continuous at the point u=φ (x 0 ). Then the complex function f(φ (x)), consisting of continuous functions, is continuous at the point x 0 .
Theorem 3. If the function y=f(x) is continuous and strictly monotone on [ a; b] axis Ohthen inverse function at=φ (x) is also continuous and monotone on the corresponding segment [ c;d] axis OU.
Any elementary function is continuous at every point at which it is defined.
5. Properties of functions continuous on a segment
Weierstrass' theorem. If a function is continuous on a segment, then it reaches its maximum and minimum values \u200b\u200bon this segment.
Consequence. If a function is continuous on a segment, then it is bounded on the segment.
Bolzano-Cauchy theorem.If the function y=f(x) is continuous on the segment [ a;
b] and takes unequal values \u200b\u200bat its ends f(a)=A and f(b)=B,
, then whatever the number FROMconcluded between AND and IN,there is a point
such that f(c)=C.
Geometrically the theorem is obvious. For any number FROMconcluded between AND and IN, there is a point with inside this segment such that f(FROM)=C... Straight at=FROM will intersect the graph of the function at least one point.
Consequence. If the function y=f(x) is continuous on the segment [ a; b] and takes values \u200b\u200bof different signs at its ends, then inside the segment [ a; b] there is at least one point fromin which the function y=f(x) vanishes: f(c)=0.
Geometricmeaning of the theorem: if the graph of a continuous function passes from one side of the axis Oh to the other, then it crosses the axis Oh.
This article is about continuous numeric function. For continuous mappings in various branches of mathematics, see continuous mapping.
Continuous function - a function without "jumps", that is, one in which small changes in the argument lead to small changes in the value of the function.
A continuous function, generally speaking, is a synonym for the concept of a continuous mapping; nevertheless, most often this term is used in a narrower sense - for mappings between number spaces, for example, on the real line. This article is devoted specifically to continuous functions defined on a subset of real numbers and taking real values.
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Definition
If we "fix" the function f (\\ displaystyle f) at the point of a removable discontinuity and put f (a) \u003d lim x → a f (x) (\\ displaystyle f (a) \u003d \\ lim \\ limits _ (x \\ to a) f (x)), then you get a function that is continuous at this point. Such an operation on a function is called by extending the definition of a function to a continuous or by extending the definition of a function by continuity, which justifies the name of the point, as a point disposable break.
Breakpoint "jump"
The gap "jump" occurs if
lim x → a - 0 f (x) ≠ lim x → a + 0 f (x) (\\ displaystyle \\ lim \\ limits _ (x \\ to a-0) f (x) \\ neq \\ lim \\ limits _ (x \\ to a + 0) f (x)).Break point "pole"
A pole gap occurs if one of the one-sided limits is infinite.
lim x → a - 0 f (x) \u003d ± ∞ (\\ displaystyle \\ lim \\ limits _ (x \\ to a-0) f (x) \u003d \\ pm \\ infty) or lim x → a + 0 f (x) \u003d ± ∞ (\\ displaystyle \\ lim \\ limits _ (x \\ to a + 0) f (x) \u003d \\ pm \\ infty). [ ]The point of significant break
At the point of significant discontinuity, one of the one-sided limits is absent altogether.
Classification of isolated singular points in R n, n\u003e 1
For functions f: R n → R n (\\ displaystyle f: \\ mathbb (R) ^ (n) \\ to \\ mathbb (R) ^ (n)) and f: C → C (\\ displaystyle f: \\ mathbb (C) \\ to \\ mathbb (C)) there is no need to work with breakpoints, but often you have to work with singular points (points where the function is not defined). The classification is similar.
There is no concept of "leap". What's in R (\\ displaystyle \\ mathbb (R)) is considered a jump, in spaces of higher dimensions it is an essential singular point.
Properties
Local
- Function continuous at a point a (\\ displaystyle a), is bounded in some neighborhood of this point.
- If the function f (\\ displaystyle f) continuous at the point a (\\ displaystyle a) and f (a)\u003e 0 (\\ displaystyle f (a)\u003e 0) (or f (a)< 0 {\displaystyle f(a)<0} ), then f (x)\u003e 0 (\\ displaystyle f (x)\u003e 0) (or f (x)< 0 {\displaystyle f(x)<0} ) for all x (\\ displaystyle x)close enough to a (\\ displaystyle a).
- If functions f (\\ displaystyle f) and g (\\ displaystyle g) continuous at the point a (\\ displaystyle a), then the functions f + g (\\ displaystyle f + g) and f ⋅ g (\\ displaystyle f \\ cdot g) are also continuous at the point a (\\ displaystyle a).
- If functions f (\\ displaystyle f) and g (\\ displaystyle g) continuous at the point a (\\ displaystyle a) and wherein g (a) ≠ 0 (\\ displaystyle g (a) \\ neq 0), then the function f / g (\\ displaystyle f / g) is also continuous at the point a (\\ displaystyle a).
- If the function f (\\ displaystyle f) continuous at the point a (\\ displaystyle a) and the function g (\\ displaystyle g) continuous at the point b \u003d f (a) (\\ displaystyle b \u003d f (a)), then their composition h \u003d g ∘ f (\\ displaystyle h \u003d g \\ circ f) continuous at the point a (\\ displaystyle a).
Global
- compact set), is uniformly continuous on it.
- A function that is continuous on a segment (or any other compact set) is bounded and reaches its maximum and minimum values \u200b\u200bon it.
- Function range f (\\ displaystyle f)continuous on a segment is the segment [min f, max f], (\\ displaystyle [\\ min f, \\ \\ max f],) where the minimum and maximum are taken along the segment [a, b] (\\ displaystyle).
- If the function f (\\ displaystyle f) continuous on the segment [a, b] (\\ displaystyle) and f (a) ⋅ f (b)< 0 , {\displaystyle f(a)\cdot f(b)<0,} then there is a point at which f (ξ) \u003d 0 (\\ displaystyle f (\\ xi) \u003d 0).
- If the function f (\\ displaystyle f) continuous on the segment [a, b] (\\ displaystyle) and the number φ (\\ displaystyle \\ varphi) satisfies the inequality f (a)<
φ
<
f
(b)
{\displaystyle f(a)<\varphi
- A continuous mapping of a segment to a real line is injective if and only if the given function on the segment is strictly monotone.
- Monotone function on a segment [a, b] (\\ displaystyle) is continuous if and only if the range of its values \u200b\u200bis a segment with ends f (a) (\\ displaystyle f (a)) and f (b) (\\ displaystyle f (b)).
- If functions f (\\ displaystyle f) and g (\\ displaystyle g) continuous on the segment [a, b] (\\ displaystyle), and f (a)<
g
(a)
{\displaystyle f(a)
Examples of
Elementary functions
This function is continuous at every point x ≠ 0 (\\ displaystyle x \\ neq 0).
The point is the break point first kind, and
lim x → 0 - f (x) \u003d - 1 ≠ 1 \u003d lim x → 0 + f (x) (\\ displaystyle \\ lim \\ limits _ (x \\ to 0-) f (x) \u003d - 1 \\ neq 1 \u003d \\ lim \\ limits _ (x \\ to 0+) f (x)),while at the very point the function vanishes.
Step function
Step function defined as
f (x) \u003d (1, x ⩾ 0 0, x< 0 , x ∈ R {\displaystyle f(x)={\begin{cases}1,&x\geqslant 0\\0,&x<0\end{cases}},\quad x\in \mathbb {R} }is continuous everywhere except for the point x \u003d 0 (\\ displaystyle x \u003d 0)where the function suffers a break of the first kind. However, at the point x \u003d 0 (\\ displaystyle x \u003d 0) there is a right-hand limit that matches the value of the function at this point. Thus, this function is an example continuous right function throughout the domain.
Similarly, a step function defined as
f (x) \u003d (1, x\u003e 0 0, x ⩽ 0, x ∈ R (\\ displaystyle f (x) \u003d (\\ begin (cases) 1, & x\u003e 0 \\\\ 0, & x \\ leqslant 0 \\ end ( cases)), \\ quad x \\ in \\ mathbb (R))is an example continuous left function throughout the domain.
Dirichlet function
f (x) \u003d (1, x ∈ Q 0, x ∈ R ∖ Q (\\ displaystyle f (x) \u003d (\\ begin (cases) 1, & x \\ in \\ mathbb (Q) \\\\ 0, & x \\ in \\ In this lesson, we will learn to establish the continuity of a function. We will do this with the help of limits, and one-sided - right and left, which are not scary at all, despite the fact that they are written like and.But what is the continuity of a function in general? Until we get to a strict definition, it is easiest to imagine a line that can be drawn without lifting the pencil from the paper. If such a line is drawn, then it is continuous. This line is the graph of a continuous function.
Graphically, a function is continuous at a point if its graph does not "break" at that point. The graph of such a continuous function -
shown in the figure below. 
Determination of the continuity of a function through the limit.
The function is continuous at a point if three conditions are met: 1. The function is defined at a point.
If at least one of the listed conditions is not met, the function is not continuous at the point. In this case, they say that the function has a break, and the points on the graph at which the graph is interrupted are called the break points of the function. The graph of such a function, which is discontinuous at the point x \u003d 2, is shown in the figure below.
Example 1.
Function) is defined as follows: f(xWill this function be continuous at each of the boundary points of its branches, that is, at the points
Decision. We check all three conditions for the continuity of the function at each boundary point. The first condition is met, since the fact that x = 0 , x = 1 , x = 3 ?
function defined at each of the boundary points follows from the definition of the function. It remains to check the other two conditions. Dot
\u003d 0. Find the left-hand limit at this point: x Let's find the right-hand limit:
.
\u003d 0 must be found for that branch of the function that includes this point, that is, the second branch. We find them:
x As you can see, the limit of the function and the value of the function at the point
\u003d 0 are equal. Therefore, the function is continuous at the point x \u003d 1. Find the left-hand limit at this point: x = 0 .
\u003d 0. Find the left-hand limit at this point: x{!LANG-660d2a28eda380aa5c8ecb530c1083dc!}
\u003d 0 must be found for that branch of the function that includes this point, that is, the second branch. We find them:
Function limit and function value at point x \u003d 1 must be found for the branch of the function that includes this point, that is, the second branch. We find them:
.
Function limit and function value at point x \u003d 1 are equal. Therefore, the function is continuous at the point x = 1 .
\u003d 0. Find the left-hand limit at this point: x \u003d 3. Find the left-hand limit at this point:
\u003d 0 must be found for that branch of the function that includes this point, that is, the second branch. We find them:
Function limit and function value at point x \u003d 3 must be found for the branch of the function that includes this point, that is, the second branch. We find them:
.
Function limit and function value at point x \u003d 3 are equal. Therefore, the function is continuous at the point x = 3 .
The main conclusion: this function is continuous at each boundary point.
Set the continuity of the function at a point yourself, and then see the solution
Continuous change in a function can be defined as a gradual change without jumps, in which a small change in the argument entails a small change in the function.
Let us illustrate this continuous function change with an example.
Let a load hang over the table. Under the action of this load, the thread is stretched, so the distance l weight from the suspension point of the thread is a function of the weight of the weight m , i.e l = f(m) , m≥0 .
If we slightly change the mass of the load, then the distance l little change: small changes m small changes correspond l ... However, if the mass of the load is close to the tensile strength of the thread, then a slight increase in the weight of the load can cause the thread to break: distance l will abruptly increase and become equal to the distance from the suspension point to the table surface. Function graph l = f(m) shown in the figure. On the site, this graph is a continuous (solid) line, and at a point it is interrupted. The result is a graph with two branches. At all points, except for, the function l = f(m) is continuous, and at the point it has a discontinuity.
The study of a function for continuity can be both an independent task and one of the stages of a complete study of a function and building its graph.
Continuity of the function on the interval
Let the function y = f(x) defined in the interval] a, b[and is continuous at every point of this interval. Then it is called continuous in the interval] a, b[. The concept of continuity of a function on intervals of the form] - ∞, b[ , ]a, + ∞ [,] - ∞, + ∞ [. Now let the function y = f(x) is defined on the segment [ a, b]. Difference between interval and line: The endpoints of the interval are not included in the interval, and the endpoints of the line are included in the line. Here we should mention the so-called one-sided continuity: at the point a, remaining on the segment [ a, b], we can only approach from the right, and to the point b - only on the left. The function is called continuous on the segment [ a, b], if it is continuous at all interior points of this segment, is continuous on the right at the point a and is continuous on the left at the point b.
Any of the elementary functions can serve as an example of a continuous function. Each elementary function is continuous on any segment on which it is defined. For example, functions and are continuous on any segment [ a, b], the function is continuous on the segment [ 0 , b], the function is continuous on any segment not containing the point a = 2 .
Example 4.Examine the function for continuity.
Decision. We check the first condition. The function is not defined at points - 3 and 3. At least one of the conditions for the continuity of the function on the entire number line is not satisfied. Therefore, this function is continuous on the intervals
.Example 5. Determine at what value of the parameter a continuous throughout areas of definition function 
Decision.
Let us find the right-hand limit at:
.
Obviously, the value at the point x \u003d 2 should be equal ax :
a = 1,5 .
Example 6.Determine at what values \u200b\u200bof the parameters a and b continuous throughout areas of definition function 
Decision.
Find the left-hand limit of the function at the point:
.
Therefore, the value at the point must be 1:
Find the left-handed function at the point:
Obviously, the value of the function at the point must be equal to:
Answer: the function is continuous over the entire domain of definition at a = 1; b = -3 .
Basic properties of continuous functions
Mathematics came to the concept of a continuous function by studying, first of all, various laws of motion. Space and time are endless and addiction like paths s from time t by law s = f(t) , gives an example of a continuous function f(t). The temperature of the heated water also changes continuously, it is also a continuous function of time: T = f(t) .
In mathematical analysis, some properties have been proven that continuous functions have. Here are the most important of these properties.
1. If a continuous function on an interval takes values \u200b\u200bof different signs at the ends of the interval, then at some point of this segment it takes a value equal to zero. More formally, this property is given in a theorem known as the first Bolzano-Cauchy theorem.
2. Function f(x), continuous on the interval [ a, b], takes all intermediate values \u200b\u200bbetween the values \u200b\u200bat the endpoints, that is, between f(a) and f(b). More formally, this property is given in a theorem known as the second Bolzano-Cauchy theorem.
Continuity of function. Break points.
There is a goby, swaying, sighing on the go:
- Oh, the board ends, now I'm going to fall!
On this lesson we will analyze the concept of continuity of a function, the classification of discontinuity points and a common practical problem function continuity studies... From the very name of the topic, many intuitively guess what will be discussed, and think that the material is quite simple. It's true. But it is simple tasks that are most often punished for neglect and a superficial approach to their solution. Therefore, I recommend that you study the article very carefully and catch all the subtleties and techniques.
What do you need to know and be able to do?Not very much. For high-quality assimilation of the lesson, you need to understand what it is function limit... Readers with a low level of training just need to comprehend the article Limits of functions. Examples of solutions and see the geometric meaning of the limit in the manual Graphs and properties of elementary functions... It is also advisable to familiarize yourself with geometric transformations of graphs, since practice in most cases involves building a drawing. Prospects are optimistic for everyone, and even a full kettle will be able to cope with the task on its own in the next hour or two!
Continuity of function. Break points and their classification
Continuity of a function
Consider some function that is continuous on the whole number line: 
Or, more laconically speaking, our function is continuous on (the set of real numbers).
What is the “philistine” criterion of continuity? Obviously, the graph of a continuous function can be drawn without lifting the pencil from the paper.
In this case, one should clearly distinguish between two simple concepts: function domain and continuity of function... In general they are not the same... For instance: 
This function is defined on the whole number line, that is, for each value "x" there is a meaning "game". In particular, if, then. Note that the other point is punctured, because by the definition of the function, the value of the argument must match the only thing function value. Thus, domain our function:.
but this function is not continuous on! It is quite clear that at the point she endures break... The term is also quite intelligible and descriptive, indeed, the pencil here will have to be torn off the paper anyway. A little later, we will look at the classification of break points.
Continuity of a function at a point and on an interval
In one way or another math problem we can talk about the continuity of a function at a point, the continuity of a function on an interval, a half-interval, or the continuity of a function on a segment. I.e, there is no "just continuity" - the function can be continuous WHERE-THAT. And the fundamental building block of everything else is continuity of function at the point .
The theory of mathematical analysis gives a definition of the continuity of a function at a point with the help of "delta" and "epsilon" neighborhoods, but in practice there is another definition in use, which we will pay the closest attention to.
Let's remember first one-sided limitswho burst into our lives in the first lesson about function graphs... Consider an everyday situation: 
If you approach the axis along the point left (red arrow), then the corresponding values \u200b\u200bof the "players" will go along the axis to the point (crimson arrow). Mathematically, this fact is fixed using left-hand limit:
Pay attention to the entry (it reads "x tends to ka on the left"). "Additive" "minus zero" symbolizes , in fact, this means that we are approaching the number from the left side.
Similarly, if you approach the point "ka" on right (blue arrow), then the "games" will come to the same value, but already along the green arrow, and right-hand limit will be formalized as follows:
"Additive" symbolizes , and the entry reads like this: "x tends to ka on the right."
If the one-sided limits are finite and equal (as in our case): 
, then we will say that there is a GENERAL limit. It's simple, the general limit is our "usual" function limitequal to a finite number.
Note that if the function is not defined at (poke out a black dot on the graph branch), then the above calculations remain valid. As has been repeatedly noted, in particular, in the article on infinitesimal functions, expressions mean that "x" infinitely close approaches the point, while IRRELEVANTwhether the function itself is defined at a given point or not. A good example will be found in the next section when a function is analyzed.
Definition: a function is continuous at a point if the limit of the function at this point is equal to the value of the function at this point:.
The definition is detailed in the following conditions:
1) The function must be defined at a point, that is, a value must exist.
2) There must be an overall function limit. As noted above, this implies the existence and equality of unilateral limits: 
.
3) The limit of the function at this point must be equal to the value of the function at this point:.
If violated at least one from three conditions, then the function loses the property of continuity at the point.
Continuity of the function on the interval is formulated in a clever and very simple way: a function is continuous on an interval if it is continuous at every point of a given interval.
In particular, many functions are continuous on an infinite interval, that is, on the set of real numbers. This is a linear function, polynomials, exponential, sine, cosine, etc. And in general, any elementary function continuous on its areas of definition, for example, logarithmic function continuous on the interval. Hopefully by now you have a pretty good idea of \u200b\u200bwhat the main function graphs look like. More detailed information about their continuity can be obtained from a kind person by the name of Fichtengolts.
With the continuity of a function on a segment and half-intervals, everything is also easy, but it is more appropriate to talk about this in the lesson. on finding the minimum and maximum values \u200b\u200bof the function on the segment, but for now we will not hammer our heads.
Break point classification
The fascinating life of functions is rich in all sorts of special points, and the breaking points are just one of the pages of their biography.
Note : just in case, I will focus on an elementary moment: a breakpoint is always single point - there are no "several break points in a row", that is, there is no such thing as "break interval".
These points, in turn, are divided into two large groups: breaks of the first kind and breaks of the second kind... Each gap type has its own characteristics, which we will look at right now:
Breakpoint of the first kind
If the continuity condition is violated at a point and one-sided limits finite then it is called break point of the first kind.
Let's start with the most optimistic case. According to the initial idea of \u200b\u200bthe lesson, I wanted to tell the theory "in general", but in order to demonstrate the reality of the material, I settled on the version with specific characters.
Sadly, like a photo of the newlyweds against the background of the Eternal Flame, but the following frame is generally accepted. Let's draw a graph of the function in the drawing: 
This function is continuous on the whole number line, except for the point. Indeed, the denominator cannot be zero. However, in accordance with the meaning of the limit - we can infinitely close to approach "zero" both to the left and to the right, that is, one-sided limits exist and, obviously, coincide: 
(Condition # 2 of continuity is satisfied).
But the function is not defined at the point, therefore, the condition No. 1 of continuity is violated, and the function suffers a discontinuity at this point.
A gap of this kind (with the existing general limit) are called removable gap... Why disposable? Because the function can be redefine at the break point: 
Looks strange? Maybe. But this function does not contradict anything! Now the gap has been closed and everyone is happy: 
Let's do a formal check:
2) 
- there is a general limit;
3)
Thus, all three conditions are satisfied, and the function is continuous at the point by the definition of the continuity of the function at the point.
However, haters of matan can redefine the function in a bad way, for example 
:
It is curious that the first two conditions of continuity are fulfilled here:
1) - the function is defined at this point;
2) 
- there is a general limit.
But the third milestone is not passed:, that is, the limit of the function at the point not equal the value of this function at this point.
Thus, the function breaks at a point.
The second, sadder case is called break of the first kind with a jump... And sadness is evoked by one-sided limits that are finite and different... An example is shown in the second drawing of the lesson. Such a gap occurs, as a rule, in piecewise-defined functionsalready mentioned in the article about graph transformations.
Consider a piecewise function 
and execute its drawing. How to build a graph? Very simple. On the half-interval we draw a fragment of a parabola (green), on the interval - a straight line segment (red) and on the half-interval - a straight line (blue).
Moreover, due to inequality, the value is determined for a quadratic function (green point), and due to inequality, the value is determined for a linear function (blue point): 
In the most, most difficult case, one should resort to point-by-point construction of each piece of the graph (see the first lesson about function graphs).
Now we will only be interested in the point. Let's examine it for continuity:
2) Let's calculate the one-sided limits.
On the left we have a red line segment, so the left-sided limit is: 
On the right is the blue line, and the right-hand limit: 
As a result, received finite numbersand they not equal... Since the one-sided limits are finite and different: 
then our function suffers break of the first kind with a jump.
It is logical that the gap cannot be eliminated - the function really cannot be redefined and "not glue", as in the previous example.
Breakpoints of the second kind
Usually all other rupture cases are slyly classified in this category. I will not list everything, because in practice in 99% of tasks you will encounter endless break - when left-handed or right-handed, and more often, both limits are infinite.
And, of course, the most suggestive picture is the hyperbole at point zero. Here, both one-sided limits are infinite: 
therefore, the function suffers a discontinuity of the second kind at a point.
I try to fill my articles with as diverse content as possible, so let's take a look at a function graph that hasn't been seen yet: 
according to the standard scheme:
1) The function is not defined at this point because the denominator vanishes.
Of course, one can immediately conclude that the function suffers a break at a point, but it would be good to classify the nature of the break, which is often required by condition. For this:
I remind you that the recording means infinitesimal negative number, and under the entry - infinitesimal positive number.
The one-sided limits are infinite, which means that the function suffers a discontinuity of the 2nd kind at a point. The ordinate axis is vertical asymptote for the schedule.
It is not uncommon for both one-sided limits to exist, but only one of them is infinite, for example: 
This is the graph of the function.
Let us investigate the point for continuity:
1) The function is not defined at this point.
2) Let's calculate the one-sided limits: 
We will talk about the methodology for calculating such one-sided limits in the last two examples of the lecture, although many readers have already seen and guessed everything.
The left-side limit is finite and equal to zero (we “do not go” to the point itself), but the right-side limit is infinite and the orange branch of the graph is infinitely close to its vertical asymptotegiven by the equation (black dotted line).
So the function suffers break of the second kind at the point.
As for the break of the 1st kind, at the very point of the break, the function can be defined. For example, for a piecewise function 
feel free to put a black bold point at the origin. On the right is a branch of hyperbole, and the right-side limit is infinite. I think almost everyone has an idea of \u200b\u200bwhat this graph looks like.
What everyone was looking forward to:
How to investigate a function for continuity?
The study of the function for continuity at a point is carried out according to the already knurled routine scheme, which consists in checking three conditions of continuity:
Example 1
Explore function 
Decision:
1) The only point in which the function is not defined falls under the sight.
2) Let's calculate the one-sided limits: 
One-sided limits are finite and equal.
Thus, the function suffers a removable discontinuity at a point.
What does the graph of this function look like?
I would like to simplify 
, and it seems to be an ordinary parabola. BUT the original function is not defined at the point, so the following caveat is required:
Let's execute the drawing: 
Answer: the function is continuous on the whole number line except for the point at which it suffers a removable discontinuity.
The function can be redefined in a good or bad way, but by condition it is not required.
You say, a contrived example? Not at all. We met dozens of times in practice. Almost all the tasks of the site come from real independent and control works.
Let's get rid of our favorite modules:
Example 2
Explore function 
for continuity. Determine the nature of the function gaps, if they exist. Execute a blueprint.
Decision: for some reason, students are afraid and do not like functions with a module, although there is nothing complicated about them. We have already touched on such things a little in the lesson. Geometric transformations of graphs... Since the modulus is non-negative, it is expanded as follows: 
, where "alpha" is some expression. In this case, our function should be signed in a piecewise manner: 
But the fractions of both pieces have to be reduced by. Reduction, as in the previous example, will not go without consequences. The original function is undefined at the point, since the denominator vanishes. Therefore, the system should additionally specify a condition, and make the first inequality strict: 
Now about a VERY USEFUL solution: before finishing the task on a draft, it is beneficial to make a drawing (regardless of whether it is required by condition or not). This will help, firstly, to immediately see the points of continuity and break points, and, secondly, it will 100% save you from mistakes when finding one-sided limits.
Let's complete the drawing. In accordance with our calculations, to the left of the point it is necessary to draw a fragment of a parabola (blue), and to the right - a piece of a parabola (red), while the function is not defined at the point itself: 
If in doubt, take several "x" values, plug them into the function 
(not forgetting that the module destroys the possible minus sign) and check the graph.
Let us investigate the function for continuity analytically:
1) The function is not defined at a point, so we can immediately say that it is not continuous at it.
2) Establish the nature of the discontinuity, for this we calculate the one-sided limits: 
One-sided limits are finite and different, which means that the function suffers a discontinuity of the first kind with a jump at a point. Note again that when finding the limits, it doesn't matter whether the function is defined at the break point or not.
Now it remains to transfer the drawing from the draft (it was made, as it were, with the help of research ;-)) and complete the task:
Answer: the function is continuous on the whole number line except for the point at which it suffers a discontinuity of the first kind with a jump.
Sometimes it is required to additionally indicate the gap jump. It is calculated in an elementary way - from the right limit, you need to subtract the left limit:, that is, at the point of discontinuity, our function jumped 2 units down (as indicated by the minus sign).
Example 3
Explore function 
for continuity. Determine the nature of the function gaps, if they exist. Make a drawing.
This is an example for independent decision, a sample solution at the end of the tutorial.
Let's move on to the most popular and widespread version of the task, when the function consists of three parts:
Example 4
Examine the function for continuity and graph the function 
.
Decision: it is obvious that all three parts of the function are continuous on the corresponding intervals, so it remains to check only two points of the "joint" between the pieces. First, let's make a drawing on a draft; I commented out the construction technique in sufficient detail in the first part of the article. The only thing you need to carefully follow our special points: due to inequality, the value belongs to a straight line (green point), and due to inequality, the value belongs to a parabola (red point): 
Well, in principle, everything is clear \u003d) It remains to make a decision. For each of the two "butting" points, we standardly check 3 conditions of continuity:
I) Let us investigate the point
1) 
One-sided limits are finite and different, which means that the function suffers a discontinuity of the first kind with a jump at a point.
We calculate the discontinuity jump as the difference between the right and left limits:
, that is, the chart jumped one unit up.
II) Let us investigate the point
1) 
- the function is defined at a given point.
2) Find one-sided limits: 
- one-sided limits are finite and equal, which means that there is a common limit.
3) 
- the limit of a function at a point is equal to the value of this function at a given point.
At the final stage, we transfer the drawing to the final copy, after which we put the final chord:
Answer: the function is continuous on the whole number line, except for the point at which it suffers a discontinuity of the first kind with a jump.
Example 5
Examine the function for continuity and plot its graph 
.
This is an example for an independent solution, a short solution and an approximate example of a task at the end of the lesson
One might get the impression that at one point the function must necessarily be continuous, and at the other, there must necessarily be a discontinuity. In practice, this is not always the case. Try not to neglect the remaining examples - there will be some interesting and important chips:
Example 6
The function is given 
... Examine the function for continuity at points. Build a graph.
Decision: and again immediately execute the drawing on the draft: 
The peculiarity of this graph is that at, the piecewise function is given by the equation of the abscissa axis. Here, this section is drawn in green, and in a notebook it is usually highlighted in bold with a simple pencil. And, of course, do not forget about our rams: the value belongs to the tangent branch (red dot), and the value belongs to the straight line.
Everything is clear from the drawing - the function is continuous on the entire number line, it remains to draw up a solution, which is brought to complete automatism literally after 3-4 similar examples:
I) Let us investigate the point
1) - the function is defined at this point.
2) Let's calculate the one-sided limits: 
so there is a general limit.
For every fireman, let me remind you of a trivial fact: the limit of a constant is equal to the constant itself. In this case, the zero limit is zero itself (left-hand limit).
3) 
- the limit of a function at a point is equal to the value of this function at a given point.
Thus, a function is continuous at a point by the definition of the continuity of a function at a point.
II) Let us investigate the point
1) - the function is defined at this point.
2) Find one-sided limits: 
And here - the limit of the unit is equal to the unit itself.
- there is a general limit.
3) 
- the limit of a function at a point is equal to the value of this function at a given point.
Thus, a function is continuous at a point by the definition of the continuity of a function at a point.
As usual, after research, we transfer our drawing to a clean copy.
Answer: the function is continuous at points.
Please note that in the condition we were not asked about the study of the entire function for continuity, and it is considered good mathematical form to formulate precise and precise the answer to the question posed. By the way, if by condition it is not required to build a graph, then you have every right not to build it (however, then the teacher can force you to do it).
A small mathematical "tongue twister" for an independent solution:
Example 7
The function is given 
... Examine the function for continuity at points. Classify breakpoints, if any. Execute a blueprint.
Try to correctly "pronounce" all the "words" \u003d) And draw the graph more precisely, accuracy, it will not be superfluous everywhere ;-)
As you remember, I recommended that you immediately execute the drawing on a draft, but from time to time you come across examples where you can't immediately figure out what the graph looks like. Therefore, in a number of cases it is advantageous to first find one-sided limits and only then draw the branches on the basis of research. In the two final examples, we will also master the technique of calculating some one-sided limits:
Example 8
Examine the function for continuity and plot its schematic graph.
Decision: bad points are obvious: (turns the denominator of the indicator to zero) and (turns to zero the denominator of the whole fraction). It is not clear what the graph of this function looks like, which means that it is better to do some research first.
Definition of Heine Continuity
The real variable function \\ (f \\ left (x \\ right) \\) is said to be continuous at the point \\ (a \\ in \\ mathbb (R) \\) (\\ (\\ mathbb (R) - \\) the set of real numbers), if for any sequence \\ (\\ left \\ (((x_n)) \\ right \\) \\ \\ right) \u003d f \\ left (a \\ right). \\] In practice, it is convenient to use the following \\ (3 \\) conditions for the continuity of the function \\ (f \\ left (x \\ right) \\) at the point \\ (x \u003d a \\) ( which must be executed simultaneously):
- The function \\ (f \\ left (x \\ right) \\) is defined at the point \\ (x \u003d a \\);
- Limit \\ (\\ lim \\ limits_ (x \\ to a) f \\ left (x \\ right) \\) exists;
- The equality is \\ (\\ lim \\ limits_ (x \\ to a) f \\ left (x \\ right) \u003d f \\ left (a \\ right) \\).
Definition of Cauchy continuity (\\ (\\ varepsilon - \\ delta \\) notation)
Consider the function \\ (f \\ left (x \\ right) \\), which maps the set of real numbers \\ (\\ mathbb (R) \\) to another subset \\ (B \\) of real numbers. The function \\ (f \\ left (x \\ right) \\) is said to be continuous at the point \\ (a \\ in \\ mathbb (R) \\), if for any number \\ (\\ varepsilon\u003e 0 \\) there exists a number \\ (\\ delta\u003e 0 \\) such that for all \\ (x \\ in \\ mathbb (R) \\) satisfying the relation \\ [\\ left | (x - a) \\ right | Definition of Continuity in Terms of Argument and Function Increments
The definition of continuity can also be formulated using argument and function increments. The function is continuous at the point \\ (x \u003d a \\) if the equality \\ [\\ lim \\ limits _ (\\ Delta x \\ to 0) \\ Delta y \u003d \\ lim \\ limits _ (\\ Delta x \\ to 0) \\ left [( f \\ left ((a + \\ Delta x) \\ right) - f \\ left (a \\ right)) \\ right] \u003d 0, \\] where \\ (\\ Delta x \u003d x - a \\).
The above definitions of the continuity of a function are equivalent on the set of real numbers.
The function is continuous on a given interval if it is continuous at every point of this interval.
Continuity theorems
Theorem 1.Let the function \\ (f \\ left (x \\ right) \\) be continuous at the point \\ (x \u003d a \\) and \\ (C \\) is constant. Then the function \\ (Cf \\ left (x \\ right) \\) is also continuous for \\ (x \u003d a \\).
Theorem 2.
Given two functions \\ ((f \\ left (x \\ right)) \\) and \\ ((g \\ left (x \\ right)) \\), continuous at the point \\ (x \u003d a \\). Then the sum of these functions \\ ((f \\ left (x \\ right)) + (g \\ left (x \\ right)) \\) is also continuous at the point \\ (x \u003d a \\).
Theorem 3.
Suppose that two functions \\ ((f \\ left (x \\ right)) \\) and \\ ((g \\ left (x \\ right)) \\) are continuous at the point \\ (x \u003d a \\). Then the product of these functions \\ ((f \\ left (x \\ right)) (g \\ left (x \\ right)) \\) is also continuous at the point \\ (x \u003d a \\).
Theorem 4.
Given two functions \\ ((f \\ left (x \\ right)) \\) and \\ ((g \\ left (x \\ right)) \\), continuous for \\ (x \u003d a \\). Then the ratio of these functions \\ (\\ large \\ frac ((f \\ left (x \\ right))) ((g \\ left (x \\ right))) \\ normalsize \\) is also continuous for \\ (x \u003d a \\) provided such that \\ ((g \\ left (a \\ right)) \\ ne 0 \\).
Theorem 5.
Suppose that the function \\ ((f \\ left (x \\ right)) \\) is differentiable at the point \\ (x \u003d a \\). Then the function \\ ((f \\ left (x \\ right)) \\) is continuous at this point (i.e., differentiability implies the continuity of the function at the point; the converse is not true).
Theorem 6 (Limit Value Theorem).
If the function \\ ((f \\ left (x \\ right)) \\) is continuous on a closed and bounded interval \\ (\\ left [(a, b) \\ right] \\), then it is bounded above and below on this interval. In other words, there are numbers \\ (m \\) and \\ (M \\) such that \\ for all \\ (x \\) in the interval \\ (\\ left [(a, b) \\ right] \\) (Figure 1).
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Fig. 1 |
Fig. 2 |
Let the function \\ ((f \\ left (x \\ right)) \\) be continuous on a closed and bounded interval \\ (\\ left [(a, b) \\ right] \\). Then, if \\ (c \\) is some number greater than \\ ((f \\ left (a \\ right)) \\) and less \\ ((f \\ left (b \\ right)) \\), then there is a number \\ (( x_0) \\) such that \\ This theorem is illustrated in Figure 2.
Continuity of elementary functions
All elementary functions are continuous at any point in their domain of definition.
The function is called elementary
if it is built from a finite number of compositions and combinations
(using \\ (4 \\) actions - addition, subtraction, multiplication and division)
... Lots of basic elementary functions
includes: