Let a function f be continuous. Assume $\int_a^b f=0$ Prove $f(x)=0$ for all $x\in [a, b]$
If $f:[a,b]\to \mathbb{R}$ is continuous and nonnegative and $\int_a^b{f}=0$, then $f(x)=0$ for all $x\in [a,b]$