Any help please with the following problem?
Let $f:X\rightarrow Y$ be a continuous one-to-one and surjective mapping between compact metric spaces. Prove that the inverse mapping $f^{-1}:Y\rightarrow X$ is continuous.
I tried the following: I assumed $C$ a subset of $X$, then by compactness of $X$, C is also compact and hence it is closed. $f$ is continuous and $C$ is compact, then $f(C)$ is compact and hence it is closed. Any help how to go from here to prove that $f^{-1}$ is continuous?