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.

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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.