Continuous function

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 point that is the limit for a certain set can be determined using the concept of a limit, namely: a function must have a limit at this point, which is equal to its value at the limit point.

If these conditions are violated at some point, it is said that the function at a given point suffers a discontinuity, that is, its continuity is violated. In the language of limits, the point of discontinuity can be described as a mismatch of the value of a function at a discontinuous point with the limit of a function (if it exists).

The point of discontinuity can be eliminated, for this it is necessary to have the limit of a function, 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 extended to continuity.
A completely different picture is formed if the limit of the function at a given point does not exist. There are two possible variants of break points:

• Of the first kind - both of the one-sided limits exist and are 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 unilateral limits do not exist or their values are infinite.

Properties of continuous functions

• The function obtained in the result of arithmetic operations, as well as the superposition of continuous functions on their domain of definition, is also continuous.
• If a continuous function is given that is positive at some point, then one can always find a sufficiently small neighborhood of it, on which it retains its sign.
• Similarly, if its values at two points A and B are equal, respectively, a and b, with a different from b, then for intermediate points it will take all values from the interval (a; b). From here we can draw an interesting conclusion: if we give a stretched rubber band to shrink so that it does not sag (remained straight), then one of its points will remain fixed. And geometrically this means that there is a straight line passing through any intermediate point between A and B, which crosses the graph of the function.

We note some of the continuous (on the domain of their definition) elementary functions:

• constant;
• Rational;
• Trigonometric.

Between the two fundamental concepts in mathematics - continuity and differentiability - there is an inextricable link. It suffices only to recall that for the differentiability of a function it is necessary that this be a continuous function.

If, however, the function is differentiable at some point, then it is continuous. However, it is not necessary that its derivative be continuous either.

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

Another important concept of mathematical analysis 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 property is considered on the set of points, and not in any one taken separately.

If we fix the point, we get nothing but the definition of continuity, that is, the existence of uniform continuity implies 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 compactum, that is, on a closed interval, then it is uniformly continuous on it.