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 $f(x)=x^3+x+1$ .

• This function is continuous and strictly increasing on $\mathbb{R}$ , thus it is a bijection of $\mathbb{R}$ onto $f(\mathbb{R})$ .
• By Prop. 5.1, $\underset{x \rightarrow +\infty}{\text{lim}} f(x) =\underset{x \rightarrow +\infty}{\text{lim}} x^3 = +\infty$ and $\underset{x \rightarrow -\infty}{\text{lim}} f(x) =\underset{x \rightarrow -\infty}{\text{lim}} x^3 = -\infty$ .

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 $\mathbb{R}$ .