F(x) is defined as x = x0F(x) has a limit when x→x0; The limit value is equal to the function value

In addition, there is a proposition that the basic elementary function is continuous in its domain and the elementary function is continuous in its domain interval.

Therefore, to judge the continuity of a function, it is generally necessary to observe whether the function is an elementary function (a function composed of a basic elementary function after a finite number of four operations and compounding). If so, then every point in the defined interval is continuous!

If the function is a piecewise function, then consider the continuity of each segment first, and then consider the continuity of the segmentation points. The method adopted is judged according to the definition!

(2) The differentiability of the function mainly depends on whether the limit lim δ y/δ x = lim [f (x)-f (x0)]/(x-x0) exists.

For basic elementary functions, they are also derivable in their domain. If you encounter piecewise function, remember that the differentiability of piecewise points must be judged by definition! In addition, for unary functions, derivability must be continuous, and vice versa may not be true!

Extended data:

Properties of continuous functions

(1) What if? (x) and g(x) are continuous at x=α, then? (x) g(x)，？ (x)g(x)，？ (As long as g(α)≠0) is also continuous at x=α. ?

2 for example? (x) continuous at x=α, and? (α)≠0, which must be within a small δ neighborhood of x = α (i.e. | x-α|)

(3) A continuous function on a closed interval must have an upper bound and a lower bound, a maximum value and a minimum value, and all intermediate values between the minimum value and the maximum value can be taken. ?

It can also be proved that all elementary functions are continuous within the defined interval. ?

Let I be a closed interval or an open interval, if ε >; 0, there must be δ > 0 exists, so for any two points x, x' in I, just | x-x' |

There are the following important theorems about uniform continuity: the continuous function on a closed interval must be uniformly continuous on this interval. This theorem is sometimes called cantor theorem. ?

References:

Baidu Encyclopedia-Continuous Function