$\def\d{\mathrm{d}}$Let $f$ be integrable. I want to show there exist two functions $g$ and $h$ that are continuous under a closed interval $[a,b]$ s.t $h\leq f\leq g$ and at the same time$$\int_{a}^{b} (g(x)-h(x)) \,\d x <ε.$$
I know that because $f$ is integrable there exist two steps functions $h\leq f\leq g$ such that$$\int_{a}^{b}g(x) \,\d x - \int_{a}^{b} h(x) \,\d x <ε,$$ but I'm having trouble in the continuity part
My intuition: I'm thinking of "joining" the steps using straight lines in order to have a continuous function. but I have no idea how to formalize it.
Thanks in advance!
I think the picture to have in mind is this:
First, we need to find step functions $f_{\rm red} \leq f$ and $f_{\rm blue} \geq f$ such that $$ \int (f_{\rm blue} - f_{\rm red} )<\frac \epsilon 3.$$ This is possible because $f$ is Riemann integrable.
Next, we want to find "joined-up-step-functions" $f_{\rm green} \leq f_{\rm red}$ and $f_{\rm orange} \geq f_{\rm blue}$ such that $$ \int( f_{\rm red} - f_{\rm green}) < \frac \epsilon 3.$$ $$ \int (f_{\rm orange} - f_{\rm blue}) < \frac \epsilon 3.$$ This is easy to achieve: we just need to make the width of each "triangle" small enough that the areas of the triangles add up to less than $\frac \epsilon 3$. To spell it out: if there are $N$ intervals in the partition, then we should choose the width of each triangle to be less than $2\epsilon /3hN$, where $h$ is the height of the triangle.