Characterization of Continuous Maps in Terms of Preservation of Connected Sets

We prove that if X is locally connected and Y is locally compact Hausdorff, then f: X -> Y is continuous if and only if f has a closed graph and f(C) is connected for any connected subset C of X. This result is not new. We show by examples that the conditions on X and Y cannot be removed. We then provide an interesting example of a non-locally connected X for which our main result holds.

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