real analysis - Continuous Functions and Metric Spaces

Let $(X, d_x)$ and $(Y, d_y)$ be metric spaces. Let $f: X \rightarrow Y$ be a function.

Show that $f$ is continuous (w.r.t. the metrics $d_x, d_y$) if and only if $f$ inverts closed sets to closed sets.

I know that a continuous function maps compact sets to compact sets. Would the proof for this have something to do with that? Since compact sets are closed and bounded, it makes sense that a continuous function has to map closed sets to closed sets (for the compact sets to compact sets fact to hold).

I'm a bit stuck on the closed sets to closed sets $\implies$ continuous function part. Every point in a closed set is an accumulation point; and, for a function to be continuous, whenever a real sequence $(a_n)$ converges to $p$, then $f(a_n)$ must converge to $f(p)$ as well. How do I make the connection between the RHS and LHS?

Any help would be greatly appreciated. Thank you.