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Can the statement that $f$ is a $C^n$ function (say for $\mathbb{R}\to\mathbb{R}$ functions) be written in terms of big/small O?
Maybe it is equivalent to existence of numbers $f(a),\dots,f^{(n)}(a)$ such that one of the following holds?
$f(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \dots + \frac{f^{(n)}(a)}{n!}(x-a) ^n + O((x-a)^{n+1})$
$f(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \dots + \frac{f^{(n)}(a)}{n!}(x-a) ^n + o((x-a)^n)$
No they are both weaker; both properties can only encode the existence of the first derivative when $n=1$, but cannot got any further. Take for instance the function $$ f(x) = \exp\left(-\frac1{|x|}\right) \sin\left(\frac1{|x|}\right). $$ It satisfies both properties for every $a$ and every $n$, but $f'$ is unbounded near $0$ (hence discontinuous, because it vanishes at the origin).
