I "created" this problem approximately 10-14 hours ago.
I think that this problem could be used as some undergraduate exercise in the chapter on Riemann integrability.
The meaning of between is the following:
Suppose that $f$ and $g$ are two Riemann-integrable functions defined on the closed interval $I$ and that $f<g$.
$f<g$ means that for every $x \in I$ it is true that $f(x)<g(x)$.
The function $h$ is between $f$ and $g$ if and only if $f<h<g$.
I think that different approaches exist to solve this exercise, and, which one would be yours?
One intuitive idea is to choose for every $f$ and $g$ some $h$ which is very wildly discontinuous in such a way for Riemann integral of $h$ to not exist.
But, I didn´t turn this idea into a concrete proof yet.
How to prove this?