Example of closed and discontinuous transformation

Question

Give an example of a closed and discontinuous transformation. This task is from compact spaces section.

I have problem with this, i can't find any solution.

Answer 1

Let $f$ be the identity map from $\mathbb R$ with usual metric to $\mathbb R$ with discrete metric. Then $f$ is not continuous but it is a closed map. Another example where both spaces are compact: define $f:[0,1] \to \{0,1\}$ by $f(x)=0$ if $x <\frac 1 2$ and $f(x)=1$ otherwise. Note that image of any set is closed!.