Does this function $$f(x,y) = -\frac{2xy}{(\frac{3}{2}\sqrt{|y|}+1+x^2)} $$ Satisfy the local Lipshitz condition $$|f(x,y_2)-f(x,y_1)| \leq M|y_2-y_1| $$ for $x,y \in \mathbb{R}$
I have no idea how to approach this. If I come up with something I will post my attempt.
Actually if $x\ne 0,$ then $f_y(x,0)$ does not exist because of the pesky term $\sqrt {|y|}.$ Note that $f$ is odd in $y.$ A differentiable odd function $g$ on $\mathbb R$ always satisfies $g'(0)=0.$ But you can see from your own formula for $f_y$ away from the $x$-axis that $\lim_{y\to 0^+} f_y(x,y)$ is nonzero if $x$ is nonzero. Therefore $f_y(x,0)$ does not exist if $x$ is nonzero.
That's not a problem however. After all, on $\mathbb R$ the function $|t|$ is Lipschitz even though the derivative doesn't exist at $0.$ We have the same general phenomenon here.