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Thread: [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 $\displaystyle R^n$. The function $\displaystyle F:R^n \rightarrow R$ defined by

    $\displaystyle f(x) = \left\{ {\begin{array}{*{20}c}
    {1} & {x \in A} \\
    {0} & {x \notin A} \\
    \end{array}} \right.
    $

    is called the characteristic function of A. Show that $\displaystyle \lim_{x \to x*} f(x)$ exists if and only if x* is either an interior or exterior point of the set A.
    =====================
    Since A is a subset of $\displaystyle R^n$, then let x* be a point in $\displaystyle 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 $\displaystyle \lim_{x \to x*} = 1 $ since x* is contained in A therefore $\displaystyle x \in A$?

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

    Thank you very much for your help.

    Never mind. I figured it out.
    Last edited by Paperwings; Aug 19th 2008 at 09:43 AM.
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