Revision as of 10:37, 18 July 2008 by Rabridge (Talk)

  1. 4. Use Ho(e)lder's Inequality to obtain

$ 0\leq \int_0^xf\leq \big[ \int_0^x(f^2) ] ^\frac{1}{2} [ \int_0^x 1^2]^\frac{1}{2}= ||f\chi_{[0,x]}||_2 x^\frac{1}{2} $

Assuming $ x>0 $ we have

$ \dfrac{F(x)}{x} \leq ||f\chi_{[0,x]}||_2 \to 0 $ as $ x\to 0_+ $

since $ f^2 \in L^1 $

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