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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.

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  • $\begingroup$ This is a little bit vague, since any problem can be phrased in terms of a function. e.g., is there a function $f: \mathbb{R} \rightarrow \mathbb{R}$ so that $\lim_{x \rightarrow \infty} f(x) = \infty$ and so that $f(x)$ is less than the size of the set $\{ n < x: n \text{ is a twin prime}\}$? This is equivalent to the twin prime conjecture... $\endgroup$ – Jair Taylor Jul 3 '15 at 6:16
  • $\begingroup$ It would help if you were a lot more specific about what kind of functions you are interested in. $\endgroup$ – Jair Taylor Jul 3 '15 at 6:18
  • $\begingroup$ An earlier, related question is math.stackexchange.com/questions/238680/… $\endgroup$ – Gerry Myerson Jul 3 '15 at 7:05
  • $\begingroup$ Is really the problem with $J$ open (until I miss some point) ? if $f: \mathbb R \to \mathbb R$ is continous, then $f(\mathbb Q)$ is countable, but $f(J)$ is countable by hypothesis. It follow that $f(\mathbb R)$ is countable, connected and not reduced to a point : contradiction. $\endgroup$ – user4422 Jul 15 '15 at 9:12
  • $\begingroup$ no it is not open, but there is no such function is proveable, I asked for such questions which are open now. It was just a sample [solved] example , I put in $\endgroup$ – Bhaskar Vashishth Jul 15 '15 at 9:14
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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 $x$.

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