# Thread: Postive integrable continuous function problem

1. ## Postive integrable continuous function problem

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

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

Proof.

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

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

2. For a continuous function $f\left( {x_0 } \right) > 0 \Rightarrow \quad \left( {\exists \delta > 0} \right)\left[ {\forall x \in \left( {x_0 - \delta ,x_0 + \delta } \right):f(x) > 0} \right]
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