This question is related to my another question posted earlier, and I think if this question can be answered, there might be a chance of answering my earlier posted one.
Given a non-decreasing stepwise function $g(\alpha)=\sum_{i=1}^N a_i1_{A_i}(\alpha)$, ($1_{A_i}(\alpha)=1$ if $\alpha \in A_i$ and $0$ otherwise), which is upper semicontinuous, is it possible to find a non-decreasing $f$ such that $f(g)$ becomes continuous?