For continuity, there are many phenomena in nature, such as temperature changes and plant growth, which are constantly changing. The reflection of this phenomenon in the function relationship is the continuity of the function. To put it simply, if you can draw an image of a function with one stroke, then the whole process is continuous, and there is no need to raise a pen.

Let the function f(x) be defined in the neighborhood of point x0. If so, say that the function is at point x0? And x0 is called the continuous point of the function. The necessary and sufficient condition for the function f(x) to be continuous at the point x0 is that the function y=f(x) is continuous around the point x0.

Extended data:

1. "Discontinuity" refers to a point where three continuity conditions cannot be met at the same time:

1, the function is not defined at this point;

2. If the function is defined at this point, but the limit of the function near this point does not exist;

3. Although the function is defined at this point and the limit exists, they are not equal.

Second, the continuous function theorem:

Theorem 1 A finite function that is continuous at a certain point is still a continuous function at that point after a finite number of operations of sum, difference, product and quotient (denominator is not 0).

Theorem 2 The inverse function of a continuous monotone increasing (decreasing) function is also continuously monotone increasing (decreasing).

Theorem 3 The composite function of a continuous function is continuous.

These properties can be derived from the definition of continuity and the related properties of limit.

References:

Baidu Encyclopedia-Continuous Function