Let (f: mathbb{R}^{d} ightarrow mathbb{R}^{n}) be a bi-Lipschitz map, i.e. both (f) and (f^{-1}) are Lipschitz continuous.

Question:

Let $f: \mathbb{R}^{d} ightarrow \mathbb{R}^{n}$ be a bi-Lipschitz map, i.e. both $f$ and $f^{-1}$ are Lipschitz continuous. Show that $\operatorname{dim} f(E)=\operatorname{dim} E$. Is this also true for a Hölder continuous map with index $\gamma \in(0,1)$ ?

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