Suppose $f$ is a Riemann integrable function and for $\epsilon>0$, there is a continuous function $g$ so that $\int |f(x)-g(x)| \ dx < \epsilon$. Show that the support of $g$, $\text{supp } g$, can be taken to be an arbitrarily small neighborhood of the support of $f$, $\text{supp }f$.
Since we can make the integral of $|f-g|$ arbitrarily small, wouldn't this mean that the support of $f$ and support of $g$ are similar in size?
How can we have support of $g$ be arbitrarily small inside the support of $f$?