I am currently working through several problems regarding the following fact:
For $F \subset \mathbb{R}$, $f: F \rightarrow \mathbb{R}$, we have that $dim_H(f(F)) \leq dim_H(F)$
I am fine with this and have proved it in a couple of different ways. Right now, I am trying to find a specific example such that $dim_H(f(F)) < dim_H(F)$, provided that $f$ is differentiable (with continuous derivative).
I've tried several different examples, but have gotten nowhere. I know that $f$ will be Lipschitz, and that $f'$ will be bounded on any interval in $\mathbb{R}$, but I'm not sure if that will be of much help to be here. One of my thoughts is that I need $f$ to be Lipschitz, but not Bilipschitz, but I'm not sure if such a function would even exist?
Any hints in the right direction would be greatly appreciated!
Take $f$ to be a constant function, and then $\dim_H(f(F))=0$ for any $F$.