How to show $\dim_\mathcal{H} f(F) \leq \dim_\mathcal{H} F$ for any set $F \subset \mathbb{R}$ and $f$ continuously differentiable? Let $f: \mathbb{R} \to \mathbb{R}$ be differentiable with continuous derivative. I have to show that for all sets $F \subset \mathbb{R}$, the inequality $$\dim_\mathcal{H} f(F) \leq \dim_\mathcal{H} F$$ holds, where $\dim_\mathcal{H}$ denotes the Hausdorff dimension. For some strange reason, there seems to be no definition of the Hausdorff dimension in the provided lecture notes. I looked it up on wikipedia and don't really know how I can say anything about the Hausdorff dimension of the image of a continuously differentiable function.
Could anyone give me some help? Thanks a lot in advance.

Hint: Show that the inequality is true if $f$ is lipschitz. Then, deduce the general case from the following property: $\dim_{\mathcal{H}} \bigcup\limits_{i \geq 0} A_i= \sup\limits_{i \geq 0} \ \dim_{\mathcal{H}}A_i$.
For a reference, there is Fractal Geometry by K. Falconer.
