*Disclaimer*: Feel free to downvote or vote to close, if this is again trivial (I seem to have a bad day today; I promise that if this is again a bummer question, I will wait $\geq 1$ day before asking new questions).

For any space $(X,\tau)$, let $\text{Cont}(X,X)$ denote the set of continuous self-maps of $X$ and let $X^X$ denote the set of all self-maps of $X$, endowed with the product topology.

What is an example of a connected $T_2$-space $(X,\tau)$ such that $\text{Cont}(X,X)$ is not dense in $X^X$?