Number of continuously differentiable functions $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfying $f(x)>0$ and $f'(x)=f(f(x))$ for all $x$ belonging to the set of real numbers are?
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1Did you try anything? – preferred_anon Nov 15 '19 at 19:30
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@preferred_anon no, I haven't solved this kind of problem previously. Any hints on how to start? – Tapi Nov 15 '19 at 19:32
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@MaximilianJanisch I don't think it is. The remark needed to answer this question (that any such solution is decreasing) is directly credited to a Putnam problem, which I suspect is exactly where this question came from. – preferred_anon Nov 15 '19 at 19:37
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@Tapi your question body says $f(x) > 0$, but your title says $f'(x) > 0$. Which do you mean? – preferred_anon Nov 15 '19 at 19:38
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@preferred_anon fixed it! It's $f(x)>0$ – Tapi Nov 15 '19 at 19:39
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@MaximilianJanisch Ah! You win. – preferred_anon Nov 15 '19 at 19:42
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@preferred_anon Actually the argument used there might be wrong Edit: Nevermind, I understand it now; here is the link for other people: https://math.stackexchange.com/a/408114/631742 – Maximilian Janisch Nov 15 '19 at 19:43