Lemma 1. f is continuous iff for each closed subset C of (codomain of f), is closed in (domain of f).

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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 .

Thus, .

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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.