A continuous function is a function without “jumps,” that is, one for which the condition is satisfied: small changes to the argument are followed by small changes to the corresponding values of the function. The graph of such a function is a smooth or continuous curve.
Continuity at a point 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, they say that the function at this point suffers a break, that is, its continuity is violated. In the language of limits, the discontinuity point can be described as the mismatch of the value of the function at the discontinuous point with the limit of the function (if it exists).
The discontinuity point may be removable; for this, a function limit must exist, but it does not coincide with its value at a given point. In this case, it can be “corrected” at this point, that is, further defined to continuity.
A completely different picture develops if the limit of the function at a given point does not exist. There are two possible break points:
- of the first kind — there are both finite and one of the one-sided limits, and the value of one or both of them 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 does 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 a continuous function is given that is positive at some point, then one can always find a sufficiently small neighborhood in which it retains its sign.
- Similarly, if its values at two points A and B are equal, respectively, to a and b, and a is different from b, then for intermediate points it will take all values from the interval (a; b). An interesting conclusion can be drawn from this: if you let the stretched gum shrink so that it does not sag (remains straight), then one of its points will remain motionless. But geometrically, this means that there is a line passing through any intermediate point between A and B that intersects the graph of the function.
We note some of the continuous (in the domain of their definition) elementary functions:
- constant;
- rational;
- trigonometric.
There is an inextricable link between the two fundamental concepts in mathematics — continuity and differentiability. It is enough to recall that for differentiability of a function it is necessary that it be a continuous function.
If the function is differentiable at some point, then there it is continuous. However, it is not necessary that its derivative be continuous.
A function that has a continuous derivative on some 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 a similar function is called 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 is a property that is considered on a set of points, and not in any particular one.
If we fix a point, 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.