Functions with Continuous Upper and Lower Envelopes

Given a real-valued upper semicontinuous function h and a real-valued lower semicontinuous function g on a metric space such that (1) h ≥ g pointwise and (2) h ( x ) = g ( x ) at each isolated point of the space, it is not in general possible to find a real-valued function f whose upper envelope is h and whose lower envelope is g , even if the space is compact and dense-in-itself. The purpose of this note is to show that such an f exists in the case that both h and g are continuous, and that f can be chosen to be a Borel function.