Suppose that $f$ is a nonnegative Riemann integrable function on $[a,b]$ satisfying $f(r) = 0$ for all $r\in\mathbb{Q}\cap [a,b]$. Prove that $\int_a^b f\,dx = 0$.
Since all rational function values give $f = 0$, does $f(x) = 0$? If yes, how can I show that formally?
If no, how would I approach this proof?