closed function with image compact hausdorff is continuous

Question

I need to prove that if $f:X\rightarrow{Y}$ is closed and $Y$ is compact Hausdorff, then $f$ is continuous. I tried proving it with the pre image of a closed set, and proving it is closed, but I got stuck.

Answer 1

Let $X$ be any connected topological space and $Y$ a finite set with discrete topology. Then $Y$ is compact Hausdorff and every map $f:X\to Y$ are closed (By closed I mean sending closed set to closed set), but only constant functions are continuous.