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.
Properties of continuous functions
- The function obtained as a result of arithmetic operations, as well as the superposition of continuous functions on their domain of definition, is also continuous.
- 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.
Definition. Let the function y = f(x) be defined at the point x0 and some of its neighborhood. The function y = f(x) is called continuous at point x0, If:
1. exists
2. this limit is equal to the value of the function at point x0: 
When defining the limit, it was emphasized that f(x) may not be defined at the point x0, and if it is defined at this point, then the value of f(x0) does not participate in any way in determining the limit. When determining continuity, it is fundamental that f(x0) exists, and this value must be equal to lim f(x).
Definition. Let the function y = f(x) be defined at the point x0 and some of its neighborhood. A function f(x) is called continuous at a point x0 if for all ε>0 there is a positive number δ such that for all x in the δ-neighborhood of the point x0 (i.e. |x-x0|
Here it is taken into account that the value of the limit must be equal to f(x0), therefore, in comparison with the definition of the limit, the condition of puncture of the δ-neighborhood 0 is removed
Let us give one more (equivalent to the previous) definition in terms of increments. Let's denote Δх = x - x0; we will call this value the increment of the argument. Since x->x0, then Δx->0, i.e. Δx - b.m. (infinitesimal) quantity. Let us denote Δу = f(x)-f(x0), we will call this value the increment of the function, since |Δу| should be (for sufficiently small |Δх|) less than an arbitrary number ε>0, then Δу- is also b.m. value, therefore
Definition. Let the function y = f(x) be defined at the point x0 and some of its neighborhood. The function f(x) is called continuous at point x0, if an infinitesimal increment in the argument corresponds to an infinitesimal increment in the function.
Definition. The function f(x), which is not continuous at the point x0, called discontinuous at this point.
Definition. A function f(x) is called continuous on a set X if it is continuous at every point of this set.
Theorem on the continuity of a sum, product, quotient 
Theorem on the passage to the limit under the sign of a continuous function 
Theorem on the continuity of superposition of continuous functions
Let the function f(x) be defined on an interval and be monotonic on this interval. Then f(x) can have only discontinuity points of the first kind on this segment.
Intermediate value theorem. If the function f(x) is continuous on a segment and at two points a and b (a is less than b) takes unequal values A = f(a) ≠ B = f(b), then for any number C lying between A and B, there is a point c ∈ at which the value of the function is equal to C: f(c) = C.
Theorem on the boundedness of a continuous function on an interval. If a function f(x) is continuous on an interval, then it is bounded on this interval.
Theorem on reaching minimum and maximum values. If the function f(x) is continuous on an interval, then it reaches its lower and upper bounds on this interval.
Theorem on the continuity of the inverse function. Let the function y=f(x) be continuous and strictly increasing (decreasing) on the interval [a,b]. Then on the segment there exists an inverse function x = g(y), also monotonically increasing (decreasing) on and continuous.
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 the 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 a continuous function at any point on an infinite 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
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
These inequalities are also valid for the values x, satisfying the conditions -π/2 
. 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.
ContentWe 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 at point . Then the function is continuous at the point if
.
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
:
.
Theorem on the boundedness of a continuous function
Let the function f (x) is continuous at point x 0
. Then there is a neighborhood U (x0), on which the function is limited.
Theorem on the preservation of the sign of a continuous function
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.
Proofs of the properties are given on the page “Properties of functions continuous at a point”.
Continuity of a complex function
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
Let there be a limit of the function at , and it is equal to:
.
Here is point t 0
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, 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
The point is called removable break point, if there is a limit
,
but the function at the point is either not defined or is not equal to the limit value: .
Thus, the point of removable discontinuity is the point of discontinuity of the 1st kind, at which the jump of the function is equal to zero.
Determination of the discontinuity point of the 2nd kind
The point is called point of discontinuity of the second kind, if it is not a discontinuity point of the 1st kind. That is, if there is not at least one one-sided limit, or at least one one-sided limit at a point is equal to infinity.
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. 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
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:
at .
Inverse functions
Definition of an inverse function
Let a function have a domain of definition X and a set of values Y. And let it have the property:
for all .
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 ;
for all .
Lemma on the mutual monotonicity of direct and inverse functions
If a function is strictly increasing (decreasing), then there is an inverse function that is also strictly increasing (decreasing).
Property of symmetry of graphs of direct and inverse functions
The graphs of direct and inverse functions are symmetrical with respect to the straight line.
Theorem on the existence and continuity of an inverse function on an interval
Let the function be continuous and strictly increasing (decreasing) on the segment. Then the inverse function is defined and continuous on the segment, which strictly increases (decreases).
For an increasing function. For decreasing - .
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) = a x, with base a > 0
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) for a ≠ 1
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) at ;
at .
Logarithm
Logarithmic function, or logarithm, y = log ax, 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;
(L.2) has many meanings;
(L.3) strictly increases as , strictly decreases as ;
(L.4) at ;
at ;
(L.5) ;
(L.6) at ;
(L.7) at ;
(L.8) at ;
(L.9) at .
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",
"Natural logarithm, ln x function"
Power 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 for p > 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 ( arcctg x), are continuous in their domains of definition.
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.
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: