Ondřej Čertík
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 Nov 18 comment Derivative of big O symbol $\sin(x^2)$ is infinitely differentiable at $x=0$ and the series is just a polynomial, so for it $O'(1) = O(1)$. You might be expanding around $x=\infty$, while I am asking about $x=0$. I clarified the question about this. Nov 18 revised Derivative of big O symbol Clarify that we only deal with x=0 case Nov 18 asked Derivative of big O symbol May 9 awarded Yearling Feb 25 awarded Nice Question Jan 16 awarded Commentator Jan 16 comment Is odd continuous function differentiable at $x=0$? @N.S. Please do, I didn't want to put your solution into answers myself. Jan 16 comment Solution of functional equation $f(x/f(x)) = 1/f(x)$? Very nice! So if the function $f(x)$ is analytic, then you prove that $f(x)=1$. Assuming only that $f(x)$ is continuous, then the only possible other solutions are not analytic. That helps a lot. Jan 16 comment Solution of functional equation $f(x/f(x)) = 1/f(x)$? Right. So the domain of $g(x)$ must be an interval (e.g. not a union of two disjoint intervals). However, what if we only solve this on an interval [1, 10], let's say? I guess we would run into some contradictions with domains and ranges in the functional equation. Jan 16 comment Solution of functional equation $f(x/f(x)) = 1/f(x)$? Because $g(x)=1/x$ is also a solution of $g(g(x))$ and somehow it got eliminated. So something is not right. (Of course, it would get eliminated later anyway due to $g'(0)=1$, but that's not the point.) Jan 16 revised Solution of functional equation $f(x/f(x)) = 1/f(x)$? added 969 characters in body Jan 16 awarded Editor Jan 16 revised Solution of functional equation $f(x/f(x)) = 1/f(x)$? Grammar. Jan 16 asked Solution of functional equation $f(x/f(x)) = 1/f(x)$? Jan 16 comment Is odd continuous function differentiable at $x=0$? N.S., you are right! $\frac{f(x)}{x} = \sin\frac{1}{x^2}$ which oscillates between -1 and 1 and so the limit does not exist. So this function is not differentiable at $x=0$. Thanks! Jan 16 asked Is odd continuous function differentiable at $x=0$? Oct 25 awarded Tumbleweed Oct 10 asked How to prove Gegenbauer's addition theorem? Sep 21 awarded Scholar Sep 21 accepted How to prove asymptotic limit of an incomplete Gamma function