This question is related to that one. The question below is my original one but I thought that the answer to that question was a simple way to solve my issue. @David C. Ullrich proved me wrong. So, let me ask the question I wanted to ask originally:
Is there a way to prove that an $H^1(\mathbb{R}^+)$ function $f$, with $f(0)=0$, close to $x=0$ is $\mathcal{O}(x^{1/2})$? In other words, can we prove that there is a function $\tilde{f}$ equal to $f$ almost everywhere such that $$ \limsup_{x\to 0} \left(|x|^{-1/2} |\tilde{f}|\right)<\infty? $$ I am working on a proof involving $H^1(\mathbb{R}^+)$, and this fact would be really useful. It seems reasonable since the derivative needs to be integrable across $x=0$, and $f(0)=0$, but, like the other question, I am not exactly sure how to prove that. No need of the detailed proof, just an idea of what tool I should be using.
One would think that is is possible to integrate from 0 to $x$ @David C. Ullrich's counterexample given to that question to produce a counterexample here. However, it does not seem that trivial to do, given the definition of his function.