If f<g are upper and lower semicontinuous nondecreasing functions, is there some continuous nondecreasing function such that f<h<g?

Question

If $f,g:[0,\infty)\mapsto [0,\infty)$ and $f<g$ are upper and lower semicontinuous nondecreasing functions, is there some continuous nondecreasing function $h:[0,\infty)\mapsto [0,\infty)$ such that $f<h<g?$

First part for $h$ being continuous we know from Michael’s selection theorem for perfectly normal space (https://www.jstor.org/stable/1969615). But what about being monotone?

Answer 1

EDIT: this answer whas to the original question (now changed) where both f and g are upper semicontinuous

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Maybe I missread, but it seems to me that a continuous $h$ in general does not exist:

Suppose $$f=\left\{\begin{array}{cc} 0 & x\in[0,1)\\ 10 & x\geq 1\end{array}\right.$$ $$g=\left\{\begin{array}{cc} 1 & x\in[0,1)\\ 11 & x\geq 1\end{array}\right.$$

If $f<h<g$ then $h(x)\leq 1$ for $x\in[0,1)$ and $h(x)\geq 10$ for $x\geq 1$. So $h$ cannot be continuous.