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 $ f$ be a function defined on an interval $ I$ . Suppose that $ f$ is continuous on $ I$ and that $ f$ is strictly monotonous on $ I$ . Then $ f$ is a bijection from $ I$ onto $ f(I)$ . The inverse function $ f^{-1}$ is continuous on $ f(I)$ .

Example 5.7.2   Take $ f(x)=x^3+x+1$ . Therefore $ f(\mathbb{R})=\mathbb{R}$ and $ f$ is a bijection of $ \mathbb{R}$ onto $ \mathbb{R}$ . The graphs of the function $ f$ and of its inverse function are displayed in Figure 11. Note that the computation of an explicit formula for the function $ f^{-1}$ is very hard. We can do it using a Computer Algebra System.
Figure 11: Graphs of two inverse functions

A consequence of this is that any equation of the form $ x^3+x+ \alpha =0$ has a unique solution in $ \mathbb{R}$ .

Noah Dana-Picard 2007-12-28