Monotone and Discrete Limits of Continuous Functions

Abstract

In this note we prove that for a quite large class of topological spaces every upper semi-continuous function, which is a discrete limit of continuous functions, it is also a pointwise decreasing discrete limit of continuous functions. This question was motivated by a paper of Zbigniew Grande. He asked that whether it be true for the topology of right hand continuity on the real line. He gave a partial answer showing that under some additional condition imposed on the function the answer is affirmative.