Consider the function $f:[0,1]\rightarrow \mathbb{R}$ given by

$$f(x)= x\log(x).$$

This function is not Lipschitz continuous at zero, but apparently Hölder continuous for any $\alpha<1$, i.e. there is a constant
$C\_{\alpha}$ such that

$$\vert f(x)-f(y)\vert \le C\_{\alpha} \vert x-y \vert^{\alpha},$$

but how can one show the Hölder continuity?

Please let me know if you have any questions or remarks.