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 $\\chi_{A}(a)$ ).
随便看	examples of Gaussian primes ExamplesOfGroups examples of groups ExamplesOfGrowthOfPerturbationsInChemicalOrganizations examples of growth of perturbations in chemical organizations ExamplesOfHarmonicFunctionsOnmathbbRn examples of harmonic functions on ℝ^n ExamplesOfInfiniteProducts examples of infinite products ExamplesOfInfiniteSimpleGroups examples of infinite simple groups ExamplesOfInitialStatesInTheGameOfLife examples of initial states in the Game of Life ExamplesOfIntegrallyClosedExtensions examples of integrally closed extensions ExamplesOfKaprekarNumbers examples of Kaprekar numbers ExamplesOfkconnectedGraphs examples of k-connected graphs ExamplesOfKeithNumbers examples of Keith numbers ExamplesOfKtheoryGroups examples of K-theory groups ExamplesOfLagrangianSubmanifolds examples of Lagrangian submanifolds