I am trying to find a simple example of a function $f:[0,1]\rightarrow\mathbb{R}$ which is a pointwise limit of continuous functions, but is not Riemann integrable. I know the classical example where we build some functions $F_1,F_2,\dots$ on a cantor-like set, and then define $f_n = F_1.F_2.\dots . F_n$, and so on. But I was thinking whether there is a simpler example, one that you could present to students with no experience in Measure Theory.
Any help is welcome.