Question

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuously differentiable function such that $0<c_1 \leqslant f^{\prime}(x) \leqslant$ $c_2$ for all $x$. Show that, if $F$ is an $s$-set in $\mathbb{R}$, then $\underline{D}^s(f(F), f(x))=\underline{D}^s(F, x)$ for all $x$ in $\mathbb{R}$, with a similar result for upper densities.

   Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuously differentiable function such that $0<c_1 \leqslant f^{\prime}(x) \leqslant$ $c_2$ for all $x$. Show that, if $F$ is an $s$-set in $\mathbb{R}$, then $\underline{D}^s(f(F), f(x))=\underline{D}^s(F, x)$ for all $x$ in $\mathbb{R}$, with a similar result for upper densities.

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Fractal Geometry: Mathematical Foundations and Applications

Fractal Geometry: Mathematical Foundations and Applications

Kenneth Falconer 2nd Edition

Chapter 5, Problem 2

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Step 1

- The function $f$ is continuously differentiable and its derivative $f'$ satisfies $0 < c_1 \leq f'(x) \leq c_2$ for all $x \in \mathbb{R}$. This implies that $f$ is strictly increasing and Lipschitz continuous. Show more…

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Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuously differentiable function such that $0<c_1 \leqslant f^{\prime}(x) \leqslant$ $c_2$ for all $x$. Show that, if $F$ is an $s$-set in $\mathbb{R}$, then $\underline{D}^s(f(F), f(x))=\underline{D}^s(F, x)$ for all $x$ in $\mathbb{R}$, with a similar result for upper densities.

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