discontinuity of characteristic function

Theorem. For a subset $A$ of $\mathbb{R}^{n}$ , the set of the discontinuity (http://planetmath.org/Continuous) points of the characteristic function $\chi_{A}$ is the boundary (http://planetmath.org/BoundaryFrontier) of $A$ .

Proof. Let $a$ be a discontinuity point of $\chi_{A}$ . Then any neighborhood (http://planetmath.org/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 $\chi_{A}(a)$ ).

Title	discontinuity of characteristic function
Canonical name	DiscontinuityOfCharacteristicFunction
Date of creation	2015-02-03 21:23:33
Last modified on	2015-02-03 21:23:33
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	3
Author	pahio (2872)
Entry type	Theorem
Classification	msc 03-00
Classification	msc 26-00
Classification	msc 26A09