[SOLVED] Characteristic Function - Interior and Exterior point

**Paperwings** · August 19th 2008, 09:30 AM

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.
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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.

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