A Stone-Weierstrass theorem for certain discontinuous functions

Abstract

Let A be point separating unital subalgebra of C(T) where T is a compact metric space. For each bounded function f:T→R which is continuous on the complement of a meagre subset of T there exists a sequence (wn) of elements of the algebra A such that the sequence (wn) convergence uniformly to the function f on each compact subset of the interior of the continuity points of the function f.