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 XContinue reading “Characterization of Continuous Maps in Terms of Preservation of Connected Sets”
We present a polished version of Borel’s proof based on Cantor’s theory of sets. More specifically, we use transfinite recursion and the fact that any strictly decreasing sequence in a well-ordered set is finite in length.