Example of an identity function that's not continuous I was looking at this big list mathoverflow question about common misconceptions: < specifically as a comment to this question something brought up in the comments but not elaborated on was that the identity function is not necessarily continuous. I haven't been able to find anything about it, and I can't think of any examples for this. If anyone could offer some insight on this, I am interested to hear.

Consider $X=(\mathbb{R}, \textrm{indiscrete topology})$ and $Y=(\mathbb{R}, \textrm{euclidean topology})$. Let $\iota: X \to Y$ be the map given by $\iota(x) = x$. Observe that $\iota^{-1}(\\{z\\}) = \\{z\\}$ and $\\{z\\}$ is closed in $Y$, but not in $X$ since the only closed sets are $\emptyset, \mathbb{R}$ with the indiscrete topology.