Let $f:\mathbb R^2 \to \mathbb R^2$ be given in polar coordinates by
$\big(r,\theta\big )\mapsto \big(\psi(r),\theta+\phi(r)\big)$, for some smooth strictly increasing function $\psi:[0,\infty) \to [0,\infty)$ satisfying $\psi(0)=0$ and $\psi'(r)>0$ for all $r$, and a smooth function $\phi:(0,\infty) \to \mathbb R$.
Suppose that $\lim_{r \to 0}\phi'(r)\psi(r)$ is finite. (In particular $\lim_{r \to 0}\phi'(r) = + \infty$ or $\lim_{r \to 0}\phi'(r) = -\infty$).
Is it possible that $f$ is smooth at the origin?
To be clear, I am not asking if all such maps are smooth, but whether or not there exist such maps that are smooth at the origin.
Edit:
It would be interesting to know whether the assumption $\psi'>0$ is superfluous.
If $\psi$ goes to zero so quickly, so that all its derivatives vanish at zero, then maybe this implies that $f$ is smooth, even if $\lim_{r \to 0}\phi'(r) = \infty$.
e.g. can we say what happens when $$ \psi(x)=\begin{cases} e^{-1/x^2},&\text{if }x\ne 0\\ 0,&\text{if }x=0\;, \end{cases} $$