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Let $f:\mathbb{R}_+\rightarrow \mathbb{R}$ be a function differentiable for $x>0$ but non differentiable at $x=0$ (for instance $f=\sqrt{\cdot}$) and $g$ be a polynomial function. I know how to compute the Taylor series of $f(g(t,x))$ at a point $g(t_0,x_0)>0$. But I don't know how to construct a differentiable approximation (polynomial) when this point is $0$.

Edit: an approximation of $f$ would be a smooth function $\varphi_N$ (polynomial) such that $f(g(t,x)) = \varphi_N(t,x) + o(t^N,x^N)$.

Are there some works in this direction ?

I read here How to Approximate at Non-differentiable Point (forced Series Expansion around Branch Cut) but it doesn't investigate this point.

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  • $\begingroup$ That depends very much on the meaning that you associate to the word "approximation". $\endgroup$ Dec 29, 2019 at 22:49
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    $\begingroup$ Maybe a Pade approximation? $\endgroup$ Dec 30, 2019 at 0:10
  • $\begingroup$ Maybe Needs["FunctionApproximations`"]; MiniMaxApproximation[Sqrt[h], {h, {$MachineEpsilon, 1}, 8, 2}] $\endgroup$
    – Michael E2
    Dec 30, 2019 at 6:51
  • $\begingroup$ I am having a look with Pade approximation, I can find a Padé approximation for $f$, does the composition with $g$ will remain a approximation of$f\circ g$ ? $\endgroup$
    – Smilia
    Dec 30, 2019 at 9:08
  • $\begingroup$ PadeApproximant[Sqrt[x],{x,0,{2,2}}] just returns Sqrt[x] and the reason for that is because Mathematica can't evaluate the Series of Sqrt[x] at $0$ $\endgroup$
    – Smilia
    Dec 30, 2019 at 9:12

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