if integral of a continuous function on a closed interval is zero, so is the function itself.

let $f:[0,1]\to R$ be continuous, s.t $\int_a^b f(x)dx=0$ for $0 \le a \le b \le 1$. Show $f$ is $0$.

this is the exact text of a qualifying exam question.
I do not know if $f$ is bounded, so continuity on a bounded interval does not imply $f$ is Riemann integrable.
I am thinking of fundamental theorem of calculus but honestly I do not know how to manipulate further on. I would appreciate any help with this question.