Let $f\in L^1[0,1]$, $E_n\subset[0,1]$ be measurable subsets and $$g_n(x)=\int_0^x\chi_{E_n}(t)f(t)\mathrm{d}t$$ where $\chi_{E_n}$ is the characteristic function of the set $E_n$. Prove that there exists a subsequence $\{g_{n_k}(x)\}$ converging uniformly to a continuous function $g(x)$ on $[0,1]$.
I tried to prove that $\{g_n\}$ is equicontinuous but I can only do it when $f$ is bounded.