Theorem: If the function f is continuous at point, then there is

Note: When saving with local number, it is often taken and sometimes done.

If the functions f and g are continuous at the point, then,, are also continuous at the point.

Note: Repeat the operation of constant function y=c and function y=x for four times, and it can be concluded that polynomial function and rational function are continuous at every point in their definition domain. Similarly, from the continuity of sinx and cosx on R, it can be concluded that tanx and cotx are continuous at every point in their domain.

Theorem: If the function F is continuous at point and G is continuous at point, then the composite function is continuous at point.

Prove:

note:

For example, it is strictly monotonous and continuous in theory, so it is continuous in theory. If it is regarded as a function composed of composite functions, it is also the continuity of composite functions, which is continuous in theory.

Note: If, it is a continuous function on its defined interval.

For example, it is proved that the rational power function is continuous within its defined interval.

Certificate:

Definition: Let F be a function defined on the number set D. If it exists, it is said that F has the maximum (minimum) value on D, and it is also said that F has the maximum (minimum) value on D.

Note: the function f does not necessarily have a maximum or minimum in its domain d (even if f is bounded on d)

Lemma: If the function F is continuous on the closed interval [a, b], then F is bounded on the closed interval [a, b].

Prove:

Theorem: If the function F is continuous in the closed interval [a, b], then F has a maximum value and a minimum value in the closed interval [a, b].

Prove:

Theorem: If the function f is continuous in the closed interval [a, b] and the signs of f(a) and f(b) are different (), then the equation has roots in (a, b).

Theorem: Let the function f be continuous in the closed interval [a, b], and if it is between f(a) and f(b), let.

Note: If f is continuous on [a, b] and can be set, then f will be able to get all the values on the interval [f(a), f(b)] on [a, b], that is,

Prove:

Example: prove: if it makes

Certificate:

Example: Let F be continuous on [a, b], satisfy and prove: make.

Certificate:

If f is continuous in the interval i and is not a constant function, then the range f(I) is also an interval. If I is a closed interval [a, b] and the maximum value of f on [a, b] is m, then f([a, b])=[m, M], if f is above [a, b].

Theorem: If the function f is strictly monotonically continuous on [a, b], then the inverse function is continuous on its domain [f(a), f(b)] or [f(b), f(a)].

Prove:

Definition: Let F be a function defined on the interval I, and if it exists, it is said that F is uniformly continuous on the interval I..

Example: Prove that the function is inconsistent and continuous at (0, 1).

Certificate:

Example: The function F is defined on the interval I, and the necessary and sufficient conditions for proving that F is uniformly continuous on I are, if, then.

Certificate:

$ take \ delta _ n = { 1 \ over n}，\ exists x' _ n，x' _ n \ in I，| x' _ n-x'' _ n | \ lt { 1 \ over n}，with | f。

Example: It is proved that the interval (0, 1) is inconsistent and continuous.

Certificate:

F is continuous on the interval I: sometimes.

Note: The value of depends on

A local property of f

F is uniformly continuous on interval i.

Note: Only depends on

A global property of f

Theorem: If the function F is continuous in the closed interval [a, b], then F is uniformly continuous in [a, b].

Prove:

Example: Let the right end point of the interval be and the left end point of the interval be (finite interval or infinite interval, respectively), which proves that if F is consistent and continuous with the upper part, then F is also consistent and continuous with the upper part.

Certificate: