Is this function Hölder continuous?

Question

Define $f:[0,1]\rightarrow \mathbb{R}$ as $$f(x)=x^\alpha \int_x^1 y^{-\alpha-1}(y-x)^{-\alpha}dy, \quad x\in [0,1],$$ where $\alpha\in (0,1/2)$ is some fixed parameter.

Is $f$ Hölder continuous of any order? Somehow I have the feeling it is since it is an integral but theoretically this is not enough.

Any ideas on how to prove or disprove it?

Thanks a lot!

Answer 1

The change of variable $y=t\,x$ gives $$ f(x)=x^{-\alpha}\int_1^{1/x}t^{-\alpha-1}(t-1)^{-\alpha}\,dt. $$ As $x\to1$, $f$ behaves like $x^{-\alpha}$. We see that $f$ is$C^1$ on $(0,1]$, but is not defined at $x=0$.