Compact graph of a continuous map
= Compact graph of a continuous map
The map $x\mapsto(x,f(x))$ is continuous into the <product topology>. It maps a <compact space> $X$ onto the <graph of a function> of a continuous $f$. Hence that graph is compact, whether or not the <codomain> is a <Hausdorff space>. Projection to $X$ is its continuous inverse.