Assume f is continuouus and let A be a subset of (domain of f). Then, is a closed subset of (codomain of f), so its inverse is closed in (domain of f) by lemma 1.
Since (the latter set is closed), we have .
Assume is true for each subset A of (domain of f).
Let C be a closed subset of (codomain of f). Then, . Thus, , which implies that is a closed subset in (domain of f).
We conclude that f is continuous by lemma 1.