Lipschitz Continuity in one variable of a several variable function

Question

The question seems to be a rather harmless one, yet till now I am unable to come up with any proof or counterexample!! The question is the following.

Prove or disprove : If $F(x,y)$ is a $C^{1}(\mathbb{R}^{2})$ function such that $F$ is Lipschitz continuous in $y$-variable, then $F_{x}(x,y)$ is also Lipschitz continuous in $y$-variable.

Thank you.

Answer 1

Try $f(x,y) = \sin(xy)$ where $f_x(x,y) = y\cos(xy)$.