I am looking for some open problems regarding functions. Problems like,
Whether a function satisfying some properties say, X,Y,Z, exists or not, is unknown.
Like there is no function $f(x)$ such that $f'(x)=h(x)$ where $h(x)=0$ if $x<0$ and $h(x)=1$ if $x\geq 0$.
Or if $J=\mathbb{R} \setminus \mathbb{Q}$ denote the set of irrational numbers. There is no continuous map $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(\mathbb{Q}) \subseteq J$ and $f(J) \subseteq \mathbb{Q}$.
What are some interesting properties for which researchers are looking to find mappings that satisfies them.
At the Open Problem Garden, Andreas Rudinger has posted this question: Give a necessary and sufficient condition on the sequence $a_n$ such that the power series $\sum_{n=0}^{\infty}a_nx^n$ is bounded for all real $x$.