Let $f:[0,1]\to[0,1]$ be a continuous function such that its derivative $f'$ exists on $(0,1)$. My question is:
Q1. If $E\subset[0,1]$ is a nowhere dense closed subset, is $f(E)$ also nowhere dense in $[0,1]$?
If the answer is negative, what will happen when we additionaly assume that $f'\ge 0$ on $(0,1)$?
My question originates from the follow question:
Q2. If $g:[0,1]\to[0,1]$ is the Cantor function, can we find homeomorphisms $\varphi$ and $\psi$ both from $[0,1]$ to itself, such that $\psi\circ g\circ \varphi$ is differntiable on $(0,1)$?
If Q1 with the addtional assumption $f'\ge 0$ has a positive answer, then clearly it gives a negative answer to the second question, because for any $\varphi$ and $\psi$, $f=\psi\circ g\circ \varphi$ maps a nowhere dense closed set onto $[0,1]$. Otherwise, I am still interested in whether $\varphi$ and $\psi$ exist or not. Q2 comes from an attempt in providing a simple counter-example to this question for the case $X=Y=(0,1)$.
Update:
Thanks to Henno Brandsma's comment below, I realized to add a remark that Q1 has a positive answer when $f$ is (piecewise) $C^1$.
Thanks to the discussion with Jim Belk, I realized that my original argument on Q1 under the assumption that $f$ is $C^1$ was incorrect. The following is a corrected argument.
Denote the Lebesgue measure on $[0,1]$ by $|\cdot|$ and denote $C=\{x\in[0,1]:f'(x)=0\}$. Note that $C$ is a closed. Using the fact that for every closed subset $K$ of $[0,1]$, $$|f(K)|\le\int_K|f'(x)|dx,$$ or otherwise, we know that $f(C)$ is closed and $|f(C)|=0$, so $f(C)$, and hence $f(C\cap E)$, are closed and nowhere dense. Note that $[0,1]\setminus C$ is a disjoint union of at most countably many intervals, say $[0,1]\setminus C=\sqcup_n I_n$. Note that $f|_{\overline{I_n}}$ is homeomorphic, so $f(\overline{I_n}\cap E)$ is closed and nowhere dense. Then by Baire category theorem, $$f(E)=f(C\cap E)\cup\big(\cup_n f(\overline{I_n}\cap E)\big)$$ is nowhere dense.
Moreover, I removed another question similar to Q1 in this post, and started a new post for it.
Any hint or suggestion is appreciated. Thanks in advance.