In Falconer's book there is the following exercize:
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the function $f(x) = x^2$ and let $F \subset \mathbb{R}$. Show that $\text{dim}f(F) = \text{dim}_H F$.
This function is not lipschitz continuous or alpha holder continuous so I can't use the theorems about related to these inequalities. Moreover, if I think about the usual cantor set dilated by 2 and then sent through this function -- the distances between some points are being compressed and the distance between other points are being expanded. So its not obvious to me that the fractal dimension remains constant.