Given a strictly increasing function $f:[0,\infty)\to \mathbb{R}$, can the smoothness of $f^{-1}$ at $f(0)$ be characterized by asymptotics of $f$ at zero?
I am particularly interested in cases where $f$ itself is not differentiable at zero.
(By smoothness at at a boundary point, I mean the maximal $k$ such that there exists an extension to an open neighborhood that is $k$ times differentiable.)
For example, let $f(x):=x^{1/n}$ with $n>1$. Then $f$ is not differentiable at zero, but still, its inverse $f^{-1}(y)=y^n$ can be extended analytically to all of $\mathbb{R}$ if $n\in \mathbb{N}$ and has smoothness $\lfloor n\rfloor$ else.
However, I don't even know how to handle in generality the case where $f(x)\sim a_1 x^{1/2}+a_2 x+a_3x^{3/2}\dots$, let alone more complicated asymptotics such as $f(x) \sim x^{1/2}(a_1 \log x+a_2 +a_3 (\log x)^{-1}+\dots)$.