I am reading "Introduction to Analysis I" written by Mitsuo Sugiura.
p.193
Theorem 4.1
Let $K$ be $\mathbb{R}$ or $\mathbb{C}$. For two open sets $A, B$ in $K$, we assume $f$ is a bijective function from $A$ to $B$, and $f$ and $f^{-1}$ are continuous. If $f$ is differetiable at $x \in A$ and $f^{'}(x) \neq 0$, then $f^{-1}$ is differentiable at $y = f(x)$ and $(f^{-1})^{'}(y) = f^{'}(x)^{-1}$.
And my question is the following:
(1)
I don't think the assumption that $f$ is continuous is necessary for the proof of the theorem.
Am I correct?
(2)
I wonder if $f$ is a continuous bijective function from $A$ to $B$, $f^{-1}$ is also continuous.
and I wonder if $f^{-1}$ is a continuous bijective function from $B$ to $A$, $f$ is also continuous.
Am I correct?
Please answer.