For a given Riemann integrable function $f$ on $[a,b]$ and $\forall \epsilon \in \mathbb{R}$, is there a Riemann integrable simple function $g$ such that $|f(x)-g(x)|<\epsilon, \ x \in [a,b]$?
It is readily to find a simple function to approximate $f$, but it seems hard to prove such a simple function Riemann integrable. Actually, I do not think it is Riemann integrable.
However, it is also hard to find a counterexample.
Thanks in advance.