discontinuity of characteristic function

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

Proof. Let $a$ be a discontinuity point of $\\chi_{A}$ . Then anyneighborhood (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 $c\otin A$ , 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. thatone does not need to know the value $\\chi_{A}(a)$ ).

单词	DiscontinuityOfCharacteristicFunction
释义	discontinuity of characteristic function Theorem. For a subset $A$ of $\\mathbb{R}^{n}$ , the set of thediscontinuity (http://planetmath.org/Continuous)points of the characteristic function $\\chi_{A}$ is theboundary (http://planetmath.org/BoundaryFrontier) of $A$ . Proof. Let $a$ be a discontinuity point of $\\chi_{A}$ . Then anyneighborhood (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 $c\otin A$ , 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. thatone does not need to know the value $\\chi_{A}(a)$ ).
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