in my class on real analysis we are currently dealing with $ L^p $ spaces and I have been tackled with this problem from Folland's real analysis stating this:
If $ f $ is absolutely continuous on $ [\epsilon, 1] $ for $ 0 < \epsilon < 1 $ and we also know $ \int_{0}^{1} {x|f'(x)|^p dx} < \infty $ then we are to show that the limit $ \lim_{x \to 0} f(x) $ exists and is finite if p>2, and also that $ \frac{|f(x)|}{|logx|^{\frac{1}{2}}} \to 0 $ as $ x \to 0 $ for p=2, and if $ p<2 $ then we are to show $ \frac{|f(x)|}{x^{1-(p/2)}} \to 0 $ as $ x \to 0 $.
Now this is from Folland's real analysis dealing with $ L^p $ spaces subsection on inequalities so one of Chebychev's, Minkowski's (for integrals), or Holder's inequalities or possibly their generalizations so my problem is I have no idea what to do how to handle or use the given integral or use absolute continuity. I would certainly appreciate the help on this so thanks all for helping a novice student
Later hint, $p=2$: Let $\epsilon >0.$ For $0<x<\epsilon$ we have
$$|f(\epsilon)-f(x)| \le \int_x^\epsilon |f'(t)|\, dt \le (\int_x^\epsilon|f'(t)|^2t\, dt)^{1/2}(\int_x^\epsilon t^{-1}\, dt)^{1/2}$$ $$\le (\int_0^\epsilon|f'(t)|^2t\, dt)^{1/2}(\int_x^\epsilon t^{-1}\, dt)^{1/2}.$$
Note the first integral on the last line $\to 0$ as $\epsilon \to 0.$
Hint for the first limit:
$$|f(y) - f(x)| \le |\int_x^y f'(t)\,dt\,| \le \int_x^y |f'(t)|\,dt \le \int_x^y (|f'(t)|t^{1/p})(t^{-1/p})\,dt.$$
Use Holder on the last integral.