Will the inverse mapping also be continuous?

Question

Suppose a map $f:A\to B$ is continuous and invertible. Will the inverse map $f^{-1}:B\to A$ be always continuous also?

Answer 1

You mean « invertible » instead of « inventively », right?

If so, consider the mapping $\psi=Id: \mathscr{S} \rightarrow \mathscr{S}$, where $\mathscr{S}$ is the space of Schwartz functions on $\mathbb{R}$ (smooth functions of which all derivatives vanish at infinity faster than any polynomial growth).

The starting space is endowed with the norm $\|f\|_s=\|f\|_{\infty}+\|f’’\|_{\infty}$ and the target space is endowed with the norm $\|f\|_e=\|f\|_{\infty}$.

$\psi$ is continuous and invertible, but its inverse map is not continuous.