linear operator Hölder continuous

Question

Let $p\in(1,\infty)$ and $\beta:=1-\frac{1}{p}$ and define $Tf:[0,1]\rightarrow \mathbb R$, $x\mapsto \int_{[0,x]}fd\lambda$. Show:
$A_p:=\{\alpha \in (0,1): Tf\text{ Hölder continuous for the exponent $\alpha$ for every $f\in L^p([0,1])$}\}=(0,\beta]$.
Hint: Show $(0,\beta]\subseteq A_p$ and show $(\beta,1)\subseteq A_p^C$. Look at the functions $t\mapsto t^\gamma$ for $\gamma \in \mathbb R$ and choose $\|f\|_p$ as the Hölder constant.
I dont have a idea to prove this. Thanks for a hint. For the first inclusion I tried to show $|\int_{[0,x]}-\int_{[0,y]}|\leq \|f\|_p|x-y|^\alpha$ for $\alpha \in (0,\beta]\subseteq (0,1)$. I don't know how to simplify the expression $|\int_{[0,x]}-\int_{[0,y]}|$