discontinuity of characteristic function

Theorem.  For a subset A of n, the set of the discontinuity (http://planetmath.org/ContinuousMathworldPlanetmathPlanetmath) points of the characteristic functionMathworldPlanetmathPlanetmathPlanetmath χA is the boundary (http://planetmath.org/BoundaryFrontier) of A.

Proof.  Let a be a discontinuity point of χA.  Then any neighborhoodMathworldPlanetmath (http://planetmath.org/Neighborhood) of a contains the points b and c such that  χA(b)=1  and  χA(c)=0.  Thus  bA  and  cA,  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  χA(b)=1  and  χA(c)=0.  This means that χA cannot be continuous at the point a (N.B. that one does not need to know the value χA(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