Suppose that the function $\displaystyle f:[0,1] \rightarrow \mathbb {R} $ is continuous and that $\displaystyle f(x) \geq 0 \ \ \ \ \ \forall x \in [0,1]$.

Prove that $\displaystyle \int ^1 _0 f > 0$ iff there is a point $\displaystyle x_0 \in [0,1] $ at which $\displaystyle f(x_0) > 0 $

Proof.

Suppose that $\displaystyle \int ^1 _0 f > 0$, then suppose to the contrary that $\displaystyle f(x)=0 \ \ \ \forall x \in [0,1] $. But then the integral would equal to zero. Contradiction.

Conversely, suppose that $\displaystyle f(x_0) > 0 $ for some $\displaystyle x_0 \in [0,1] $... How should I continue? Thanks.