Example of $f:\mathbb{R}\to\mathbb{R}$ injective and bounded, but with inverse not bounded or injective.

Question

I am trying to come up with an example of a bounded and injective function $f:\mathbb{R}\to\mathbb{R}$ such that $f^{-1}$ is not injective or bounded.

What are examples that could apply in this situation?

Thanks for helping! :D

Answer 1

As a first thought, I would say $f^{-1}$ is always injective, since a function is invertible if and only if it is injective and surjective. $f^{-1}$ is certainly invertible since its inverse is $f$, thus it is also injective.

For the bounded question, consider the function $$ f(x) = \left\{ \begin{aligned} &\dfrac{1}{1+(\frac{1}{x})^2} &&: x > 0\\ &0 &&: x=0 \\ &-f(-x) &&: x<0 \end{aligned} \right.$$ This function is bounded between -1 and 1 and one-to-one on $\Bbb R$. It has an inverse, at least for its image (which is not all of $\Bbb R$, since it's bounded), $f^{-1}: (-1,1) \longrightarrow \Bbb R$. If you look at the graph of $f$, you can see that this inverse will not be bounded.