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Thread: Postive integrable continuous function problem

  1. #1
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    Postive integrable continuous function problem

    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.
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  2. #2
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    For a continuous function $\displaystyle 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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