Prove that if integral of a squared function is zero, then function is zero function

To complete the proof along the lines you started: $F$ is increasing (non-decreasing) on the entire interval. $F(a) = F(b)$. For any $x$ between $a$ and $b$, $F(a) \le F(x) \le F(b) = F(a)$. So the inequalities must be equalities.