general topology - Discontinuous maps taking compacts to compacts

It's commonly known that in general topology, a continuous map $f$ from a topological space $(X, \tau)$ to another topological space $(Y, \tau')$ will send every compact set to another compact set. Moreover, I'm aware that the converse does not hold, as so long as $\# Y \geq 2$, we can pick some set $S \subseteq X$ and define
$$f(x) = \begin{cases} y_{1} & x \in S \\ y_{2} & x \in S^{\complement} . \end{cases}$$
This would send not only compact sets to compact sets, but all sets to compact sets because $f(X)$ is finite, and thus compact (as are all its subsets). But this map could easily not be continuous.

NOTE: Understand for the rest of this post that $f: X \to Y$ is surjective.

My question is if there are more interesting counter-examples. Say $X = \mathbb{R}$ with the Euclidean metric; then what would be an example of such a discontinuous map $f: X \to Y$ where $Y = \mathbb{N}$ (with the usual metric), or $Y = \mathbb{Q}$ (again with the usual topology). What about $\mathbb{R}$, or $\mathbb{R} \setminus \mathbb{Q}$? Is there perhaps a limit on how large in cardinality $Y$ can be (with respect to $X$)? I'm curious to see less trivial counterexamples to the converse. Thanks!

EDIT: Many of the examples I'm getting still rely on compact sets having finite image. So a problem I might find more interesting: Can I construct a discontinuous counterexample where $f$ is also injective?