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Prove that a metric space X is discrete if and only if every function from X to an arbitrary metric space is continuous.

The attempt at a solution.

I didn't have problems to prove the implication discrete metric implies continuity. Let f:(X,δ)→(Y,d) where (Y,d) is an arbitrary metric space. Let V be an open set in (Y,d). Every set in X is open so, in particular, f^-1(V) is an open set. This proves f is continuous.

I don't know how to prove the other statement: if every function from the metric space X to some arbitrary metric space Y is a continuous function, then X is discrete.