Find an example of a map $f:(X,d_{X})\rightarrow (Y,d_{Y})$ between metric spaces such that $X$ is compact, $f$ is not continuous but $\Gamma(f):=\lbrace (x,f(x))\in X\times Y\mid x\in X\rbrace$ is closed in $X\times Y$
I though about this example:
$$f:([0,1],d_{e})\rightarrow ([0,1],d_{disc})$$
$$f(x)=x$$
Obviously, $f$ is not continuous because we have the discrete metric and $[0,1]$ is compact with the euclidean metric. So $\Gamma(f)$ is the bisector of first and third quadrant. Now, I have some problems to see if $\Gamma(f)$ is closed. I though about the fact that the discrete topology is stronger that the euclidean Topology so this bisector is closed. But I have some doubt because the product topology is particular in this case. Can someone help me? Thanks before!