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Math Help - Postive integrable continuous function problem

  1. #1
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    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.
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  2. #2
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    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]<br />
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