# Characteristic function proof

Let A be a subset of R^n. Show that the characteristic function Xa is continuous on the interior of A and on the interior of its complement A' but is discontinuous on the boundary ∂A = A (bar)∩A' (bar)

© BrainMass Inc. brainmass.com October 9, 2019, 10:27 pm ad1c9bdddfhttps://brainmass.com/math/geometry-and-topology/characteristic-function-proof-220729

#### Solution Preview

Let f: = chi_A, the characteristic function of A, with codomain {0, 1} equipped with the discrete topology.

Now, recall that a function f : X -> Y is continuous at a point x of X if, for all open V in Y, there is ...

#### Solution Summary

This provides an example of proving continuity and discontinuity of a characteristic function.

$2.19