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Homediscontinuity of characteristic function

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Theorem. For a subset $A$ of $\mathbb{R}^{n}$, the set of the
discontinuity
points of the characteristic function^{} $\chi_{A}$ is the
boundary of $A$.

Proof. Let $a$ be a discontinuity point of $\chi_{A}$. Then any neighborhood of $a$ contains the points $b$ and $c$ such that $\chi_{A}(b)=1$ and $\chi_{A}(c)=0$. Thus $b\in A$ and $c\notin A$, whence $a$ is a boundary point of $A$.

If, on the contrary, $a$ is a boundary point of $A$ and $U(a)$ an arbitrary neighborhood of $a$, it follows that $U(a)$ contains both points belonging to $A$ and points not belonging to $A$. So we have in $U(a)$ the points $b$ and $c$ such that $\chi_{A}(b)=1$ and $\chi_{A}(c)=0$. This means that $\chi_{A}$ cannot be continuous at the point $a$ (N.B. that one does not need to know the value $\chi_{A}(a)$).

## Mathematics Subject Classification

03-00*no label found*26-00

*no label found*26A09

*no label found*

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## Comments

## autolinking; "related" box

In the old PM system, the autolinking has functionned very well, but now it does not work for many usual terms. For example, in the new entry http://planetmath.org/discontinuityofcharacteristicfunction I had to use three forced links for getting the linkings (discontinuity, boundary, neighborhood

^{}). I have thought that the autolinking has been one of the best features of PM.BTW, can the old ”Related” box be added to the new PM system? Now its lacking is a defect here.