- 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^{\frac{1}{2}} \leq ||f\chi_{[0,x]}||_2 \to 0 $ as $ x\to 0^+ $
by absoulute continuity, since $ f^2 \in L^1. $
-Matty
