Let $X$ and $Y$ be topological spaces. Is there a function $f:X\to Y$ that is not continuous and it is not a closed map but its graph $Gr(f)$ is a closed subspace of $X\times Y$?
If I think about a function that is not continuous and whose graph is closed, then the only candidate I can think of is $f:S^1\to[0,2\pi),$ $f(\cos(t),\sin(t))=t$. The problem is that $f$ is a closed map.
Of course, $Y$ cannot be compact, because in that case $Gr(f)$ is closed if and only if $f$ is continuous.
Let $X=Y=\mathbb{R}$ and set $$f(x) = \begin{cases} 1/x, & x \ne 0 \\ 0, & x=0. \end{cases}$$
Then $f$ has closed graph, it is not continuous, and it is not a closed map (for instance, $[1,\infty)$ is closed but $f([1,\infty)) = (0,1]$ is not).