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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
S.
[Please see attached PDF file for full problem].

for part (a), I think something similar to
http://planetmath.org/encyclopedia/CharacteristicFunction.html
can be used, correct?

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Characteristic function of a metric space is investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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Proof:
(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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