If $f_x(0,0)$ doesn't exists, can we conclude that $f_{xy}(0,0)$ doesn't exist?

Question

Given that $f:\mathbb{R}^2 \rightarrow \mathbb{R}$. Suppose we have $f_x$ is defined for all points $(x,y)$ except $(0,0)$.

Is it possible for $f_{xy}(0,0)$ to exist?

I think it is not possible by using definition of partial derivative.

Remark: By $f_{xy}$, I mean differentiate $f$ w.r.t. $x$ first, followed by $y$.