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Math Help - [SOLVED] Characteristic Function - Interior and Exterior point

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    [SOLVED] Characteristic Function - Interior and Exterior point

    Problem:
    Let A be a subset of R^n. The function F:R^n \rightarrow R defined by

    f(x) = \left\{ {\begin{array}{*{20}c}<br />
   {1} & {x \in A}  \\<br />
   {0} & {x \notin A}  \\<br />
\end{array}} \right.<br />

    is called the characteristic function of A. Show that \lim_{x \to x*} f(x) exists if and only if x* is either an interior or exterior point of the set A.
    =====================
    Since A is a subset of R^n, then let x* be a point in R^n .

    If x* is an interior point then there is an open ball about x* that is contained in A.

    My question is that if x* is an interior point, then the \lim_{x \to x*} = 1 since x* is contained in A therefore x \in A?

    Similarly, if x* is an exterior point(provided that there is an open ball about x* that is contained in R^n\A , then \lim_{x \to x*} = 0 since x \notin A

    Thank you very much for your help.

    Never mind. I figured it out.
    Last edited by Paperwings; August 19th 2008 at 10:43 AM.
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