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    Characteristic function proof

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    Let A be a subset of R^n. Show that the characteristic function Xa is continuous on the interior of A and on the interior of its complement A' but is discontinuous on the boundary ∂A = A (bar)∩A' (bar)

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    Solution Preview

    Let f: = chi_A, the characteristic function of A, with codomain {0, 1} equipped with the discrete topology.

    Now, recall that a function f : X -> Y is continuous at a point x of X if, for all open V in Y, there is ...

    Solution Summary

    This provides an example of proving continuity and discontinuity of a characteristic function.