Continuity of functions. Continuity of functions - theorems and properties How to find intervals of continuity of a function example
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 condition for the continuity of the function y=f(x) at point X 0 is that:
Because 
, then equality (32) can be written in the form
(33)
This means that when finding the limit of a continuous functionf(x) one can go to the limit under the function sign, i.e. into a function f(x) instead of an argument X substitute its limit value X 0 .
lim sin x=sin(lim x);
lim arctan x=arctg(lim x); (34)
lim log x=log(lim x).
Exercise. Find the limit: 1) ; 2) 
.
Let us define the continuity of a function, based on the concepts of increment of argument and function.
Because conditions and 
are identical (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 a point X 0 and its neighborhood, and an infinitesimal increment in the argument corresponds to an infinitesimal increment in the function.
Exercise. Examine the continuity of a function y=2X 2 1.
Properties of functions continuous at a point
1. If the functions f(x) And φ
(x) are continuous at the point X 0, then their sum 
, work 
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)>0, then there is such 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= φ (x) is continuous at the point u 0 = φ (x 0 ), then a complex function y=f[φ (x)] is continuous at the point X 0 .
2. Continuity of a function in an interval and on a segment
Function y=f(x) is called continuous in the interval (a; b), if it is continuous at every 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=A is continuous on the right (i.e.), and at the point x=b is left continuous (i.e. 
).
3. Function discontinuity points and their classification
The points at which the continuity of a function is broken are called break points this function.
If X=X 0 – function break point y=f(x), then at least one of the conditions of the first definition of continuity of a function is not satisfied.
Example.
1.
. 2.
3)
4)
.
▼Break point X 0 is called the break point first kind functions y=f(x), if at this point there are finite limits of the function on the left and on the right (one-sided limits), i.e. 
And 
. Wherein:
Magnitude | A 1 -A 2 | called function jump at the point of discontinuity of the first kind. ▲
▼Break point X 0 is called the break point second kind functions y=f(x), if at least one of the one-sided limits (left or right) does not exist or is equal to infinity. ▲
Exercise. Find break points and find out their type for functions:
1)
;
2)
.
4. Basic theorems on continuous functions
Theorems on the continuity of functions follow directly from the corresponding theorems on limits.
Theorem 1. The sum, product and quotient of two continuous functions is a continuous function (for the quotient, except for those values of the argument in which the divisor is not equal to 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] axes Oh, then the inverse function at=φ (x) is also continuous and monotonic on the corresponding segment [ c;d] axes OU.
Every elementary function is continuous at every point at which it is defined.
5. Properties of functions continuous on an interval
Weierstrass's theorem. If a function is continuous on a segment, then it reaches its maximum and minimum values on this segment.
Consequence. If a function is continuous on an interval, then it is bounded on the interval.
Bolzano-Cauchy theorem. If the function y=f(x) is continuous on the interval [ a;
b] and takes unequal values at its ends f(a)=A And f(b)=B,
, then whatever the number is WITH, concluded between A And IN, there is a point
such that f(c)=C.
Geometrically the theorem is obvious. For any number WITH, concluded between A And IN, there is a point c inside this segment such that f(WITH)=C. Straight at=WITH intersects the graph of the function at at least one point.
Consequence. If the function y=f(x) is continuous on the interval [ a; b] and takes on the values of different signs at its ends, then inside the segment [ a; b] there is at least one point With, in which the function y=f(x) goes to zero: f(c)=0.
Geometric the meaning of the theorem: if the graph of a continuous function passes from one side of the axis Oh to the other, then it intersects the axis Oh.
Definition. The function f(x), defined in the neighborhood of some point x 0, is called continuous at a point x 0 if the limit of the function and its value at this point are equal, i.e.
The same fact can be written differently: 
Definition. If the function f(x) is defined in some neighborhood of the point x 0, but is not continuous at the point x 0 itself, then it is called explosive function, and the point x 0 is the discontinuity point.
Example of a continuous function:
y
0 x 0 - x 0 x 0 + x
P 
example of a discontinuous function:
Definition. The function f(x) is called continuous at the point x 0 if for any positive number >0 there is a number >0 such that for any x satisfying the condition
inequality true 
.
Definition. The function f(x) is called continuous at the point x = x 0, if the increment of the function at the point x 0 is an infinitesimal value.
f(x) = f(x 0) + (x)
where (x) is infinitesimal at xx 0.
Properties of continuous functions.
1) The sum, difference and product of functions continuous at the point x 0 is a function continuous at the point x 0.
2) Quotient of two continuous functions 
– is a continuous function provided that g(x) is not equal to zero at point x 0.
3) Superposition of continuous functions is a continuous function.
This property can be written as follows:
If u = f(x), v = g(x) are continuous functions at the point x = x 0, then the function v = g(f(x)) is also a continuous function at this point.
The validity of the above properties can be easily proven using limit theorems.
Continuity of some elementary functions.
1) The function f(x) = C, C = const is a continuous function over the entire domain of definition.
2) Rational function 
is continuous for all values of x except those at which the denominator becomes zero. Thus, a function of this type is continuous over the entire domain of definition.
3) Trigonometric functions sin and cos are continuous in their domain of definition.
Let us prove property 3 for the function y = sinx.
Let us write the increment of the function y = sin(x + x) – sinx, or after transformation:
Indeed, there is a limit for the product of two functions 
And 
. In this case, the cosine function is a limited function atх0 
, and because
limit of the sine function 
, then it is infinitesimal atх0.
Thus, there is a product of a bounded function and an infinitesimal one, therefore this product, i.e. function у is infinitesimal. In accordance with the definitions discussed above, the function y = sinx is a continuous function for any value x = x 0 from the domain of definition, because its increment at this point is an infinitesimal value.
Break points and their classification.
Let's consider some function f(x), continuous in the neighborhood of the point x 0, with the possible exception of this point itself. From the definition of a break point of a function it follows that x = x 0 is a break point if the function is not defined at this point or is not continuous at it.
It should also be noted that the continuity of a function can be one-sided. Let us explain this as follows.
, then the function is said to be right continuous.
If the one-sided limit (see above) 
, then the function is said to be left continuous.
Definition. The point x 0 is called break point function f(x), if f(x) is not defined at the point x 0 or is not continuous at this point.
Definition. The point x 0 is called discontinuity point of the 1st kind, if at this point the function f(x) has finite, but not equal, left and right limits.
To satisfy the conditions of this definition, it is not necessary that the function be defined at the point x = x 0, it is enough that it is defined to the left and to the right of it.
From the definition we can conclude that at the discontinuity point of the 1st kind a function can only have a finite jump. In some special cases, the discontinuity point of the 1st kind is also sometimes called removable breaking point, but we’ll talk more about this below.
Definition. The point x 0 is called point of discontinuity of the 2nd kind, if at this point the function f(x) does not have at least one of the one-sided limits or at least one of them is infinite.
Continuity of a function on an interval and on a segment.
Definition. The function f(x) is called continuous on an interval (segment), if it is continuous at any point of the interval (segment).
In this case, the continuity of the function at the ends of the segment or interval is not required; only one-sided continuity is required at the ends of the segment or interval.
Properties of functions continuous on an interval.
Property 1: (Weierstrass's first theorem (Carl Weierstrass (1815-1897) - German mathematician)). A function that is continuous on an interval is bounded on this interval, i.e. the condition –M f(x) M is satisfied on the segment.
The proof of this property is based on the fact that a function that is continuous at the point x 0 is bounded in a certain neighborhood of it, and if you divide the segment into an infinite number of segments that are “contracted” to the point x 0, then a certain neighborhood of the point x 0 is formed.
Property 2: A function that is continuous on the segment takes the largest and smallest values on it.
Those. there are values x 1 and x 2 such that f(x 1) = m, f(x 2) = M, and
m f(x) M
Let us note these largest and smallest values the function can take on a segment several times (for example, f(x) = sinx).
The difference between the largest and smallest value of a function on a segment is called hesitation functions on a segment.
Property 3: (Second Bolzano–Cauchy theorem). A function that is continuous on the interval takes on all values between two arbitrary values on this interval.
Property 4: If the function f(x) is continuous at the point x = x 0, then there is some neighborhood of the point x 0 in which the function retains its sign.
Property 5: (First theorem of Bolzano (1781-1848) - Cauchy). If a function f(x) is continuous on a segment and has values of opposite signs at the ends of the segment, then there is a point inside this segment where f(x) = 0.
Those. if sign(f(a)) sign(f(b)), then x 0: f(x 0) = 0.
Example.
at the point x = -1 the function is continuous at the point x = 1 discontinuity point of the 1st kind
at
Example. Examine the function for continuity and determine the type of discontinuity points, if any.
at the point x = 0 the function is continuous at the point x = 1 discontinuity point of the 1st kind
The definition of continuity of a function at a point is given. Equivalent definitions according to Heine, according to Cauchy and in terms of increments are considered. Determination of one-sided continuity at the ends of a segment. Formulation of lack of continuity. Examples are analyzed in which it is necessary to prove the continuity of a function using the Heine and Cauchy definitions.
ContentSee also: Limit of a function - definitions, theorems and properties
Continuity at a point
Determining the continuity of a function at a point
Function f (x) called continuous at point x 0
neighborhood U (x0) this point, and if the limit as x tends to x 0
exists and is equal to the value of the function at x 0
:
.
This implies that x 0 - this is the end point. The function value in it can only be a finite number.
Definition of continuity on the right (left)
Function f (x) called continuous on the right (left) at point x 0
, if it is defined on some right-sided (left-sided) neighborhood of this point, and if the right (left) limit at the point x 0
equal to the function value at x 0
:
.
Examples
Example 1
Using the Heine and Cauchy definitions, prove that the function is continuous for all x.
Let there be an arbitrary number. Let us prove that the given function is continuous at the point.
The function is defined for all x .
Therefore, it is defined at a point and in any of its neighborhoods.
.
We use Heine's definition
.
Let's use . Let there be an arbitrary sequence converging to: .
Applying the property of the limit of a product of sequences we have:
Since there is an arbitrary sequence converging to , then
Continuity has been proven.
We use the Cauchy definition .
Let's use .
.
Let's consider the case.
;
We have the right to consider the function on any neighborhood of the point. .
Therefore we will assume that
;
(A1.1) .
.
Let's apply the formula:
.
Taking into account (A1.1), we make the following estimate:
.
.
.
(A1.2)
Applying (A1.2), we estimate the absolute value of the difference:
(A1.3)
According to the properties of inequalities, if (A1.3) is satisfied, if and if , then .
Now let's look at the point.
In this case
We have the right to consider the function on any neighborhood of the point.
Therefore we will assume that .
Let's use .
(A2.1) .
(A2.2)
.
Let's put it.
.
Then
.
Taking into account (A2.1), we make the following estimate:
.
Then
So, .
Applying this inequality and using (A2.2), we estimate the difference:
.
(A2.3)
We introduce positive numbers and , connecting them with the following relations:
.
(A1.2)
According to the properties of inequalities, if (A2.3) is satisfied, if and if , then .
.
This means that for any positive there is always a .
.
Then for all x satisfying the inequality, the following inequality is automatically satisfied:
.
Now let's look at the point.
We need to show that the given function is continuous at this point on the right. In this case
Enter positive numbers and :
This shows that for any positive there is always .
Then for all x such that , the following inequality holds:
It means that . That is, the function is continuous on the right at the point.
O.I. Besov. Lectures on mathematical analysis. Part 1. Moscow, 2004.
L.D. Kudryavtsev. Course of mathematical analysis. Volume 1. Moscow, 2003. CM. Nikolsky. Course of mathematical analysis. Volume 1. Moscow, 1983.
See also:
Definition. Let a function f(x) be defined on some interval and x 0 be a point in this interval. If , then f(x) is said to be continuous at the point x 0 .
From the definition it follows that we can talk about continuity only in relation to those points at which f(x) is defined (when defining the limit of a function, such a condition was not set). For continuous functions
, that is, the operations f and lim are commutable. Accordingly, two definitions of the limit of a function at a point can be given two definitions of continuity - “in the language of sequences” and “in the language of inequalities” (in the language of ε-δ). It is suggested that you do this yourself.
For practical use, it is sometimes more convenient to define continuity in the language of increments. 
The value Δx=x-x 0 is called the increment of the argument, and Δy=f(x)-f(x 0) is the increment of the function when moving from point x 0 to point x. 
.
Let f(x) be defined at point x 0 . A function f(x) is called continuous at a point x 0 if an infinitesimal increment of the argument at this point corresponds to an infinitesimal increment of the function, that is, Δy→0 for Δx→0.
. The function y=f(x) is called continuous at the point x 0 on the right (left) if
.
A function continuous at an interior point will be both right and left continuous. The converse is also true: if a function is continuous at a point on the left and right, then it will be continuous at that point. However, a function can only be continuous on one side. For example, for 
, 
, f(1)=1, therefore, this function is continuous only on the left (for the graph of this function, see above in paragraph 5.7.2).
Definition.
A function is called continuous on some interval if it is continuous at every point of this interval.
In particular, if the interval is a segment, then one-sided continuity is implied at its ends.
Properties of continuous functions
1. All elementary functions are continuous in their domain of definition.2. If f(x) and φ(x), given on a certain interval, are continuous at the point x 0 of this interval, then the functions will also be continuous at this point.
3. If y=f(x) is continuous at the point x 0 from X, and z=φ(y) is continuous at the corresponding point y 0 =f(x 0) from Y, then the complex function z=φ(f(x )) will be continuous at the point x 0 .
Function breaks and their classification
A sign of continuity of the function f(x) at the point x 0 is the equality, which implies the presence of three conditions:1) f(x) is defined at point x 0 ;
2)
;
3) .
If at least one of these requirements is violated, then x 0 is called the break point of the function. In other words, a break point is a point at which this function is not continuous. From the definition of breakpoints it follows that the breakpoints of a function are:
a) points belonging to the domain of definition of the function at which f(x) loses the property of continuity,
b) points not belonging to the domain of definition of f(x), which are adjacent points of two intervals of the domain of definition of the function.
For example, for a function, the point x=0 is a break point, since the function at this point is not defined, and the function
has a discontinuity at the point x=1, which is adjacent to two intervals (-∞,1) and (1,∞) of the domain of definition of f(x) and does not exist. The following classification is adopted for break points.
1) If at the point x 0 there are finite 
And 
, but f(x 0 +0)≠f(x 0 -0), then x 0 is called discontinuity point of the first kind
, and is called function jump
.
Example 2.
Consider the function 
The function can only be broken at the point x=2 (at other points it is continuous like any polynomial). 
We'll find 
, 
. Since the one-sided limits are finite, but not equal to each other, then at the point x=2 the function has a discontinuity of the first kind. notice, that 
, therefore the function at this point is continuous on the right (Fig. 2).
2) Discontinuity points of the second kind
are called points at which at least one of the one-sided limits is equal to ∞ or does not exist.
Example 3.
The function y=2 1/ x is continuous for all values of x except x=0. Let's find one-sided limits: 
, 
, therefore x=0 is a discontinuity point of the second kind (Fig. 3).
3) Point x=x 0 is called removable break point
, if f(x 0 +0)=f(x 0 -0)≠f(x 0).
We will “eliminate” the gap in the sense that it is enough to change (redefine or redefine) the value of the function at this point by setting , and the function will become continuous at the point x 0 . 
Example 4.
It is known that 
, and this limit does not depend on the way x tends to zero. But the function at point x=0 is not defined. If we redefine the function by setting f(0)=1, then it turns out to be continuous at this point (at other points it is continuous as the quotient of the continuous functions sinx and x). 
Example 5.
Examine the continuity of a function 
.
, that is, the operations f and lim are commutable. Accordingly, two definitions of the limit of a function at a point can be given two definitions of continuity - “in the language of sequences” and “in the language of inequalities” (in the language of ε-δ). It is suggested that you do this yourself.
The functions y=x 3 and y=2x are defined and continuous everywhere, including in the indicated intervals. Let's examine the junction point of the intervals x=0: 
, 
, . We obtain that , which implies that at the point x=0 the function is continuous.
Definition.
A function that is continuous on an interval except for a finite number of points of discontinuity of the first kind or removable discontinuity is called piecewise continuous on this interval.
Examples of discontinuous functions
Example 1.
The function is defined and continuous on (-∞,+∞) except at the point x=2. Let's determine the type of break. Because the 
And 
, then at the point x=2 there is a discontinuity of the second kind (Fig. 6). 
Example 2.
The function is defined and continuous for all x except x=0, where the denominator is zero. Let's find one-sided limits at the point x=0: One-sided limits are finite and different, therefore, x=0 is a discontinuity point of the first kind (Fig. 7).
Example 3.
Determine at what points and what kind of discontinuities the function has 
This function is defined on [-2,2]. Since x 2 and 1/x are continuous in the intervals [-2,0] and , respectively, the discontinuity can only occur at the junction of the intervals, that is, at the point x=0. Since , then x=0 is a discontinuity point of the second kind.
Example 4.
Is it possible to eliminate function gaps:
A) 
at point x=2;
b) 
at point x=2;
V) 
at point x=1?
, that is, the operations f and lim are commutable. Accordingly, two definitions of the limit of a function at a point can be given two definitions of continuity - “in the language of sequences” and “in the language of inequalities” (in the language of ε-δ). It is suggested that you do this yourself.
Regarding example a) we can immediately say that the discontinuity f(x) at the point x=2 cannot be eliminated, since at this point there are infinite one-sided limits (see example 1).
b) The function g(x) although has finite one-sided limits at the point x=2
(
,
),
but they do not coincide, so the gap cannot be eliminated either.
c) The function φ(x) at the discontinuity point x=1 has equal one-sided finite limits: . Therefore, the gap can be eliminated by redefining the function at x=1 by putting f(1)=1 instead of f(1)=2.
Example No. 5. Show that the Dirichlet function 
discontinuous at every point on the numerical axis.
, that is, the operations f and lim are commutable. Accordingly, two definitions of the limit of a function at a point can be given two definitions of continuity - “in the language of sequences” and “in the language of inequalities” (in the language of ε-δ). It is suggested that you do this yourself.
Let x 0 be any point from (-∞,+∞). In any of its neighborhoods there are both rational and irrational points. This means that in any neighborhood of x 0 the function will have values equal to 0 and 1. In this case, the limit of the function at the point x 0 cannot exist either on the left or on the right, which means that the Dirichlet function has discontinuities of the second kind at each point on the real axis.
Example 6. Find function breakpoints
and determine their type.
, that is, the operations f and lim are commutable. Accordingly, two definitions of the limit of a function at a point can be given two definitions of continuity - “in the language of sequences” and “in the language of inequalities” (in the language of ε-δ). It is suggested that you do this yourself. Points suspected of breaking are points x 1 =2, x 2 =5, x 3 =3.
At the point x 1 =2 f(x) has a discontinuity of the second kind, since
.
The point x 2 =5 is a point of continuity, since the value of the function at this point and in its vicinity is determined by the second line, and not the first: .
Let's examine the point x 3 =3: ,
, from which it follows that x=3 is a discontinuity point of the first kind. For independent decision.
Examine functions for continuity and determine the type of discontinuity points:
1) 
; Answer: x=-1 – point of removable discontinuity;
2) 
; Answer: Discontinuity of the second kind at point x=8;
3) 
; Answer: Discontinuity of the first kind at x=1;
4) 
Answer: At the point x 1 =-5 there is a removable gap, at x 2 =1 there is a gap of the second kind and at the point x 3 =0 there is a gap of the first kind.
5) How should the number A be chosen so that the function 
would be continuous at x=0?
Answer: A=2.
6) Is it possible to choose the number A so that the function 
would be continuous at x=2?
Answer: no.
Definition of continuity according to Heine
The function of a real variable \(f\left(x \right)\) is said to be continuous at the point \(a \in \mathbb(R)\) (\(\mathbb(R)-\)set of real numbers), if for any sequence \(\left\( ((x_n)) \right\)\ ), such that \[\lim\limits_(n \to \infty ) (x_n) = a,\] the relation \[\lim\limits_(n \to \infty ) f\left(((x_n)) \right) = f\left(a \right).\] In practice, it is convenient to use the following \(3\) conditions for the continuity of the function \(f\left(x \right)\) at the point \(x = a\) ( which must be executed simultaneously):
- The function \(f\left(x \right)\) is defined at the point \(x = a\);
- The limit \(\lim\limits_(x \to a) f\left(x \right)\) exists;
- The equality \(\lim\limits_(x \to a) f\left(x \right) = f\left(a \right)\) holds.
Definition of Cauchy continuity (notation \(\varepsilon - \delta\))
Consider a function \(f\left(x \right)\) that maps the set of real numbers \(\mathbb(R)\) to another subset \(B\) of the real numbers. The function \(f\left(x \right)\) is said to be continuous at the point \(a \in \mathbb(R)\), if for any number \(\varepsilon > 0\) there is a number \(\delta > 0\) such that for all \(x \in \mathbb (R)\), satisfying the relation \[\left| (x - a) \right| Definition of continuity in terms of increments of argument and function
The definition of continuity can also be formulated using increments of argument and function. The function is continuous at the point \(x = a\) if the equality \[\lim\limits_(\Delta x \to 0) \Delta y = \lim\limits_(\Delta x \to 0) \left[ ( f\left((a + \Delta x) \right) - f\left(a \right)) \right] = 0,\] where \(\Delta x = x - a\).
The above definitions of 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 \(x = a\) and \(C\) be a constant. Then the function \(Cf\left(x \right)\) is also continuous for \(x = a\).
Theorem 2.
Given two functions \((f\left(x \right))\) and \((g\left(x \right))\), continuous at the point \(x = a\). Then the sum of these functions \((f\left(x \right)) + (g\left(x \right))\) is also continuous at the point \(x = a\).
Theorem 3.
Suppose that two functions \((f\left(x \right))\) and \((g\left(x \right))\) are continuous at the point \(x = a\). Then the product of these functions \((f\left(x \right)) (g\left(x \right))\) is also continuous at the point \(x = a\).
Theorem 4.
Given two functions \((f\left(x \right))\) and \((g\left(x \right))\), continuous for \(x = a\). Then the ratio of these functions \(\large\frac((f\left(x \right)))((g\left(x \right)))\normalsize\) is also continuous for \(x = a\) subject to , that \((g\left(a \right)) \ne 0\).
Theorem 5.
Suppose that the function \((f\left(x \right))\) is differentiable at the point \(x = a\). Then the function \((f\left(x \right))\) is continuous at this point (i.e., differentiability implies continuity of the function at the point; the converse is not true).
Theorem 6 (Limit value theorem).
If a 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).
|
||
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 than \((f\left(b \right))\), then there exists 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 \(4\) operations - addition, subtraction, multiplication and division)
. A bunch of basic elementary functions
includes:
