A Darboux integral

Question

let $f:[a,b] \rightarrow R$ , proof by Darboux sums: f is integrable if and only if for every $ε>0$ there exist functions $h,g:[a,b] \rightarrow$ R, that are integrable such that for exery x $\in[a,b]$: $$h(x)\leq f(x) \leq g(x)$$ and: $$\int_a^b (g(x)-h(x))<ε$$

Answer 1

Hint: For a given partition $\mathcal{P}=\{a=a_0<\ldots<x_n=b$} define $$\begin{align} h(x)&=\sum^n_{k=1}\big(\inf_{t\in[x_{j-1},x_j]}f(t)\big)\mathbb{1}_{[x_{j-1},x_j]}(x)\\ g(x)&=\sum^n_{k=1}\big(\sup_{t\in[x_{j-1},x_j]}f(t)\big)\mathbb{1}_{[x_{j-1},x_j]}(x) \end{align} $$ where $\mathbb{1}_A(x)$ stands for the function that is $1$ if $x\in A$ and $0$ otherwise. Clearly $h\leq f\leq g$. The functions $h$ and $g$ are step functions and clearly Riemann integrable. Can you finish from this?