Continuous function with compact domain has continuous inverse

To my knowledge, the most expeditious way to prove this is to use

$f$ is continuous on $X$ if and only if for any open set $V \subset Y$, $f^{-1}(V)$ is an open set in $X$.

where $X$ and $Y$ are domain and codomain of $f$ (can be arbitrary metric space).

Based on this statement, to show $f^{-1}$ is a continuous mapping of $B$ onto $A$, it is sufficient to show that for any open set $V \subset A$, $(f^{-1})^{-1}(V)$ is an open set in $B$, that is, $f(V)$ is open in $B$.

Now since $V$ is open in $A$ and $A$ is compact, $V^c \cap A$, as a closed subset of $A$, is compact (use the theorem that any closed subset of a compact set is compact). Since $f$ is continuous, $f(V^c \cap A)$ is compact and so is closed in $B$ (use the theorem that continuous mapping of any compact set is compact). Now since $f$ is one-to-one and onto, $f(V)$ is the complement of $f(V^c \cap A)$, hence is open in $B$.

For the counterexample, consider $A = [0, 1] \cup (2, 3]$, which is not compact, and define $f$ as follows: \begin{align} f(x) = \begin{cases} x & x \in [0, 1] \\ x - 1 & x \in (2, 3] \end{cases} \end{align} $f$ is a one-to-one and onto function of $A$ to $B = [0, 2]$, in addition, $f$ is continuous at every point of $A$. It is easily to get that \begin{align} f^{-1}(x) = \begin{cases} x & x \in [0, 1] \\ x + 1 & x \in (1, 2] \end{cases} \end{align} Hence $f^{-1}$ is not continuous at $1$! (a jump occurs here).

What you have:

• $f$ is bijective $A\to B$
• $\forall x\in A ~\forall \epsilon > 0 ~\exists \delta > 0 ~\forall x' \in A (|x - x'| < \delta \implies |f(x) - f(x') | < \epsilon)$.

What you want:

• $\forall x \in A~ \forall \epsilon > 0 ~\exists \delta > 0 ~\forall x' \in A (|f(x) - f(x')| < \delta \implies |x - x'| < \epsilon)$.

Proof by contradiction (sketch): assume for contradiction

$$ \exists x\in A ~ \exists \epsilon > 0 ~\forall \delta > 0 ~\exists x' \in A ~(|f(x) - f(x')| < \delta ~\wedge ~ |x - x'| \geq \epsilon )$$

Then letting $\delta_k = \frac{1}{k}$ we can find a sequence $\{x_k\}$ such that $|f(x) - f(x_k)| < 1/k$ while $|x - x_k| \geq \epsilon$.

By the Heine-Borel we have that $A$ is bounded. By Bolzano-Weierstrass we have that $\{x_k\}$ has a converging subsequence which we call $\{y_k\}$. Since $A$ is closed the limit, which we call $y$, is in $A$.

Since $f$ is continuous and $y_k \to y$ you have $f(y_k) \to f(y)$. On the other hand since by assumption $f(x_k) \to f(x)$, the subsequence $f(y_k)$ must also converge to $f(x)$. This implies $f(x) = f(y)$. But we know that $|y - x| \geq \epsilon$ by triangle inequality and so $y \neq x$. This contradicts injectivity.

I am assuming since you already have Heine-Borel you also have Bolzano-Weierstrass.