# Invertible functions.

Please recall the definition of a function, either increasing or decreasing on an interval (v.s. def monotonous function).

Theorem 5.7.1   Let be a function defined on an interval . Suppose that is continuous on and that is strictly monotonous on . Then is a bijection from onto . The inverse function is continuous on .

Example 5.7.2   Take .
• This function is continuous and strictly increasing on , thus it is a bijection of onto .
• By Prop. 5.1, and .
Therefore and is a bijection of onto . The graphs of the function and of its inverse function are displayed in Figure 11. Note that the computation of an explicit formula for the function is very hard. We can do it using a Computer Algebra System.

A consequence of this is that any equation of the form has a unique solution in .

Subsections
Noah Dana-Picard 2007-12-28