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    Characteristic Function of Metric Space

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    Let S ⊂ M.
    (a) Define the characteristic function Xs : M --> R.
    (b) If M is a metric space, show that Xs(x) is discontinuous at x if and only if x is a boundary point of
    [Please see attached PDF file for full problem].

    for part (a), I think something similar to
    can be used, correct?

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    (a) The characteristic function is defined as
    if ; if
    (b) Now suppose is a metric space. We should know what is a boundary point of . If is a boundary point of , then in each neighborhood of , we can find some , such that . In another word, any neighborhood of a ...

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