Let $\alpha\in(-1/2,0)$ and $x\in (0,1)$. Define the function $$f(x) = \int_x^1 z^{\alpha-\frac{1}{2}} (z-x)^{\alpha}dz.$$
I have the feeling that $|f(x)-f(y)| \leq C|x-y|^{1/2}$ or any other power between 0 and 1. The (improper) integral is convergent so one should expect that the resulting function is continuous of some Hölder degree. Does anyone see a quick way to prove it?
Thank you for any hint!