Continuity of a function at a point definition. Continuity of a function at a point and on an interval. With examples. Properties and continuity of elementary functions

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.

Content

See 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 equal to the value functions in 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's prove that 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
.
.


.
absolute value

differences: (A1.3) According to the properties of inequalities, if (A1.3) is satisfied, if and if , then .

Example 2

Using prove that the function is continuous for all .

The given function is defined at .

Let us prove that it is continuous at the point.
Let's consider the case.
We have the right to consider the function on any neighborhood of the point. .

Let's use .
Therefore we will assume that .
(A2.1)
.

(A2.2)


.
Let's put it.
.

Then

.
Let's put it.
Taking into account (A2.1), we make the following estimate: .

So, Applying this inequality and using (A2.2), we estimate the difference:(A2.3)
.
Enter

positive numbers
.
absolute value

and , connecting them with the relations:
.
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.

In a similar way, one can prove that the function , where n is a natural number, is continuous for . References: 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.

In a similar way, one can prove that the function , where n is a natural number, is continuous for . See also: 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:

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

In a similar way, one can prove that the function , where n is a natural number, is continuous for . function, and the point x 0 is the discontinuity point.

Example of a continuous function:
.

In a similar way, one can prove that the function , where n is a natural number, is continuous for . y 0 x 0 - x 0 x 0 + x P

example of a discontinuous function:

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

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)

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 atх0
, and because

limit of the sine function
, then it is infinitesimal atх0.

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.

In a similar way, one can prove that the function , where n is a natural number, is continuous for . 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.

In a similar way, one can prove that the function , where n is a natural number, is continuous for . 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.

In a similar way, one can prove that the function , where n is a natural number, is continuous for . 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.

In a similar way, one can prove that the function , where n is a natural number, is continuous for . 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

Let the point a belongs to the function specification area f(x) and any ε -neighborhood of a point a contains different from a points of the function definition area f(x), i.e. dot a is the limit point of the set (x), on which the function is specified f(x).

Definition. Function f(x) called continuous at a point a, if function f(x) has at point a limit and this limit is equal to the particular value f(a) functions f(x) at the point a.

From this definition we have the following function continuity condition f(x) at the point a :

Since , then we can write

Therefore, for a continuous line at a point a functions the limit transition symbol and the symbol f function characteristics can be swapped.

Definition. Function f(x) is called continuous on the right (left) at the point a, if the right (left) limit of this function at the point a exists and is equal to the private value f(a) functions f(x) at the point a.

The fact that the function f(x) continuous at a point a on the right write it like this:

And the continuity of the function f(x) at the point a on the left is written as:

Comment. Points at which a function does not have the property of continuity are called discontinuity points of this function.

Theorem. Let functions be given on the same set f(x) And g(x), continuous at a point a. Then the functions f(x)+g(x), f(x)-g(x), f(x) g(x) And f(x)/g(x)- continuous at a point a(in the case of a private one, you need to additionally require g(a) ≠ 0).

Continuity of basic elementary functions

1) Power function y=xn with natural n continuous on the entire number line.

First let's look at the function f(x)=x. By the first definition of the limit of a function at a point a take any sequence (xn), converging to a, then the corresponding sequence of function values (f(x n)=x n) will also converge to a, that is , that is, the function f(x)=x continuous at any point on the number line.

Now consider the function f(x)=x n, Where n is a natural number, then f(x)=x · x · … · x. Let's go to the limit at x → a, we get , that is, the function f(x)=x n continuous on the number line.

2) Exponential function.

Exponential function y=a x at a>1 is continuous function at any point on an infinite straight line.

Exponential function y=a x at a>1 satisfies the conditions:

3) Logarithmic function.

The logarithmic function is continuous and increasing along the entire half-line x>0 at a>1 and is continuous and decreases along the entire half-line x>0 at 0, and

4) Hyperbolic functions.

The following functions are called hyperbolic functions:

From the definition of hyperbolic functions it follows that the hyperbolic cosine, hyperbolic sine and hyperbolic tangent are defined on the entire numerical axis, and the hyperbolic cotangent is defined everywhere on the numerical axis, with the exception of the point x=0.

Hyperbolic functions are continuous at every point of their domain (this follows from the continuity of the exponential function and the theorem on arithmetic operations).

5) Power function

Power function y=x α =a α log a x continuous at every point of the open half-line x>0.

6) Trigonometric functions.

Functions sin x And cos x continuous at every point x an infinite straight line. Function y=tan x (kπ-π/2,kπ+π/2), and the function y=ctg x continuous on each interval ((k-1)π,kπ)(everywhere here k- any integer, i.e. k=0, ±1, ±2, …).

7) Inverse trigonometric functions.

Functions y=arcsin x And y=arccos x continuous on the segment [-1, 1] . Functions y=arctg x And y=arcctg x continuous on an infinite line.

Two wonderful limits

Theorem. Function limit (sin x)/x at the point x=0 exists and is equal to one, i.e.

This limit is called the first remarkable limit.

Proof. At 0 the inequalities are valid 0<\sin x. Let us divide these inequalities into sin x, then we get

These inequalities are also valid for the values x, satisfying the conditions -π/2 . This follows from the fact that cos x=cos(-x) And . Because cos x is a continuous function, then . Thus, for functions cos x, 1 and in some δ -neighborhood of a point x=0 all conditions of the theorems are satisfied. Hence, .

Theorem. Function limit at x → ∞ exists and is equal to the number e:

This limit is called second remarkable limit.

Comment. It is also true that

Continuity of a complex function

Theorem. Let the function x=φ(t) continuous at a point a, and the function y=f(x) continuous at a point b=φ(a). Then the complex function y=f[φ(t)]=F(t) continuous at a point a.

Let x=φ(t) And y=f(x)- the simplest elementary functions, with many values (x) functions x=φ(t) is the scope of the function y=f(x). As we know, elementary functions are continuous at every point of the given domain. Therefore, according to the previous theorem, the complex function y=f(φ(t)), that is, the superposition of two elementary functions, is continuous. For example, a function is continuous at any point x ≠ 0, as a complex function of two elementary functions x=t -1 And y=sin x. Also function y=ln sin x continuous at any point in the intervals (2kπ,(2k+1)π), k ∈ Z (sin x>0).

Definitions and formulations of the main theorems and properties of a continuous function of one variable are given. The properties of a continuous function at a point, on a segment, the limit and continuity of a complex function, and the classification of discontinuity points are considered. Definitions and theorems related to the inverse function are given. The properties of elementary functions are outlined.

Content

We can formulate the concept of continuity in in terms of increments. To do this, we introduce a new variable, which is called the increment of the variable x at the point.
.
Then the function is continuous at the point if
.
Let's introduce a new function: They call her function increment
.

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 :
.

at point .
Then the function is continuous at the point if (x) Theorem on the boundedness of a continuous function 0 Let the function f (x0) is continuous at point x

.
Let the function be continuous at the point.
.
And let it have a positive (negative) value at this point:
Then there is a neighborhood of the point where the function has a positive (negative) value:

at .
Arithmetic properties of continuous functions
Let the functions and be continuous at the point .
Then the functions , and are continuous at the point .

If , then the function is continuous at the point .
Left-right continuity property

A function is continuous at a point if and only if it is continuous on the right and left.

Continuity of a complex function

Proofs of the properties are given on the page “Properties of functions continuous at a point”.
Continuity theorem for a complex function
Let the function be continuous at the point.

And let the function be continuous at the point.

Then the complex function is continuous at the point.
Limit of a complex function
.
Theorem on the limit of a continuous function of a function 0 Let there be a limit of the function at , and it is equal to:
Here is point t
can be finite or infinitely distant: .
.

And let the function be continuous at the point.
Then there is a limit of a complex function, and it is equal to:
Theorem on the limit of a complex function
Let the function have a limit and map a punctured neighborhood of a point onto a punctured neighborhood of a point.
.

Let the function be defined on this neighborhood and have a limit on it.

Here are the final or infinitely distant points: .
Neighborhoods and their corresponding limits can be either two-sided or one-sided. Then there is a limit of a complex function and it is equal to: Break points
Determining the break point
Let the function be defined on some punctured neighborhood of the point .

The point is called
function break point , if one of two conditions is met: 1) not defined in ;
.

2) is defined at , but is not at this point.
Determination of the discontinuity point of the 1st kind The point is called
.

discontinuity point of the first kind
function break point , if is a break point and there are finite one-sided limits on the left and right: Definition of a function jump
,
Jump Δ function

at a point is the difference between the limits on the right and left

Determining the break point
function break point removable break point, if there is a limit

Properties of functions continuous on an interval

Definition of a function continuous on an interval
A function is called continuous on an interval (at) if it is continuous at all points of the open interval (at) and at points a and b, respectively.

Weierstrass's first theorem on the boundedness of a function continuous on an interval
If a function is continuous on an interval, then it is bounded on this interval.

Determining the attainability of the maximum (minimum)
A function reaches its maximum (minimum) on the set if there is an argument for which
for all .

Determining the reachability of the upper (lower) face
A function reaches its upper (lower) bound on the set if there is an argument for which
.

Weierstrass's second theorem on the maximum and minimum of a continuous function
A function continuous on a segment reaches its upper and lower bounds on it or, which is the same, reaches its maximum and minimum on the segment.

Bolzano-Cauchy intermediate value theorem
Let the function be continuous on the segment.
.

And let C be an arbitrary number located between the values ​​of the function at the ends of the segment: and .
Then there is a point for which
.

Corollary 1
Let the function be continuous on the segment.
Then there is a neighborhood of the point where the function has a positive (negative) value:

And let the function values ​​at the ends of the segment have different signs: or .

Then there is a point at which the value of the function is equal to zero:
Corollary 2
for all .
Let the function be continuous on the segment. Let it go . Then the function takes on the interval all the values ​​from and only these values: Inverse functions
.

Definition of an inverse function
;
Let a function have a domain of definition X and a set of values ​​Y.
for all .

And let it have the property:
Then for any element from the set Y one can associate only one element of the set X for which .

This correspondence defines a function called
inverse function

To . The inverse function is denoted as follows:
From the definition it follows that

for all ;

Theorem on the existence and continuity of an inverse function on an interval
Let the function be continuous and strictly increasing (decreasing) on ​​an open finite or infinite interval.

Then the inverse function is defined and continuous on the interval, which strictly increases (decreases).
For an increasing function.

For decreasing: .

In a similar way, we can formulate the theorem on the existence and continuity of the inverse function on a half-interval.

Properties and continuity of elementary functions

Elementary functions and their inverses are continuous in their domain of definition. Below we present the formulations of the corresponding theorems and provide links to their proofs.

Exponential function Exponential function f(x) = ax > 0 , with base a
,
is the limit of the sequence
.

where is an arbitrary sequence of rational numbers tending to x:
Theorem. Properties of the Exponential Function
The exponential function has the following properties:(P.0)
defined, for , for all ;(P.1) 1 for a ≠
has many meanings;(P.2)
strictly increases at , strictly decreases at , is constant at ; ;
(P.3) ;
(P.3*) ;
(P.4) ;
(P.5) ;
(P.6) ;
(P.7)(P.8)
continuous for all;(P.9)
Then there is a neighborhood of the point where the function has a positive (negative) value:

at ;

Logarithm Logarithmic function, or logarithm, y= log a x, with base a

is the inverse of the exponential function with base a.
Theorem. Properties of the logarithm Logarithmic function with base a, y = log a x
, has the following properties:(L.1)
defined and continuous, for and , for positive values ​​of the argument; for a ≠
(L.2)(L.3)
strictly increases as , strictly decreases as ;(P.9)
(P.9)
(L.4) ;
(L.5)(P.9)
(L.6)(P.9)
(L.7)(P.9)
(L.8) Then there is a neighborhood of the point where the function has a positive (negative) value:

(L.9)

Exponent and natural logarithm
.
In the definitions of the exponential function and the logarithm, a constant appears, which is called the base of the power or the base of the logarithm. In mathematical analysis, in the vast majority of cases, simpler calculations are obtained if the number e is used as the basis:

An exponential function with base e is called an exponent: , and a logarithm with base e is called a natural logarithm: .
The properties of the exponent and the natural logarithm are presented on the pages
"Exponent, e to the power of x",

Power function

"Natural logarithm, ln x function" Power function with exponent p is the function f(x) = xp
, the value of which at point x is equal to the value of the exponential function with base x at point p. In addition, f(0) = 0 p = 0 0 .

Here we will consider the properties of the power function y = x p for non-negative values ​​of the argument.
For rationals, for odd m, the power function is also defined for negative x.

In this case, its properties can be obtained using even or odd.
These cases are discussed in detail and illustrated on the page “Power function, its properties and graphs”.
Theorem. Properties of the power function (x ≥ 0) A power function, y = x p, with exponent p has the following properties:
(C.1)
defined and continuous on the set

at ,

at ".
Trigonometric functions Theorem on the continuity of trigonometric functions Trigonometric functions: sine ( sin x), cosine ( cos x), tangent ( tg x

) and cotangent (
ctg x Theorem on the continuity of inverse trigonometric functions Inverse trigonometric functions: arcsine ( arcsin x), arc cosine ( arccos x), arctangent ( arctan x) and arc tangent (

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.

arcctg x

), are continuous in their domains of definition.

A continuous function is a function without “jumps”, that is, one for which the condition is satisfied: small changes in the argument are followed by small changes in the corresponding values ​​of the function. The graph of such a function is a smooth or continuous curve.

Continuity at a limit point for a certain set can be defined using the concept of a limit, namely: a function must have a limit at this point that is equal to its value at the limit point.
If these conditions are violated at a certain point, they say that the function at this point suffers a discontinuity, that is, its continuity is violated. In the language of limits, a break point can be described as a discrepancy between the value of a function at the break point and the limit of the function (if it exists).

  • The break point can be removable; for this, the existence of a limit of the function is necessary, but it does not coincide with its value at a given point. In this case, it can be “corrected” at this point, that is, it can be further defined to continuity.
  • A completely different picture emerges if there is a limit to the given function. There are two possible breakpoint options:

of the first kind - both of the one-sided limits are available and finite, and the value of one of them or both does not coincide with the value of the function at a given point;

  • of the second kind, when one or both of the one-sided limits do not exist or their values ​​are infinite.
  • If you are given a continuous function that is positive at some point, then you can always find a sufficiently small neighborhood of it where it will retain its sign.
  • Similarly, if its values ​​at two points A and B are equal to a and b, respectively, and a is different from b, then for intermediate points it will take all values ​​from the interval (a ; b). From this we can draw an interesting conclusion: if you let a stretched elastic band compress so that it does not sag (remains straight), then one of its points will remain motionless. And geometrically, this means that there is a straight line passing through any intermediate point between A and B that intersects the graph of the function.

Let us note some of the continuous (in the domain of their definition) elementary functions:

  • constant;
  • rational;
  • trigonometric.

There is an inextricable connection between two fundamental concepts in mathematics - continuity and differentiability. It is enough just to remember that for a function to be differentiable it is necessary that it be a continuous function.

If a function is differentiable at some point, then it is continuous there. However, it is not at all necessary that its derivative be continuous.

A function that has a continuous derivative on a certain set belongs to a separate class of smooth functions. In other words, it is a continuously differentiable function. If the derivative has a limited number of discontinuity points (only of the first kind), then such a function is called piecewise smooth.

Another important concept is the uniform continuity of a function, that is, its ability to be equally continuous at any point in its domain of definition. Thus, this is a property that is considered at many points, and not at any one point.

If we fix a point, then we get nothing more than a definition of continuity, that is, from the presence of uniform continuity it follows that we have a continuous function. Generally speaking, the converse is not true. However, according to Cantor’s theorem, if a function is continuous on a compact set, that is, on a closed interval, then it is uniformly continuous on it.