Let $f$ be a $C^1$ function.
If $|f(x+t)-f(x)|\le |f'(x)t|+|t^n|$ for sufficiently small $t$, does it follow that $f$ is a $C^m$ function (say for $m$ strictly less than $n$, or whatever we can generalize it)?
If it does hold for all positive whole $n$, does it follow that $f$ is $C^\infty$ smooth?
Rewrite it (not quite equivalent but related):
$f(x) = f(x-c) + (x-c)f'(x-c)+o((x-c)^n)$
Then it (that $f$ is $C^n$) follows from answers to https://mathoverflow.net/q/88501
So my question is answered. But I want also a short proof for my special case.