$f(x,y) = \sin\sqrt{x^2+y^2}$

$f(x,y) = \begin{cases}\dfrac{x^3y-xy^3}{x^2+y^2}, & (x,y)\ne(0,0); \\ 0, & (x,y)=(0,0).\end{cases}$

$f(x,y) = e^{-y/4}\cos(2x)$

$f(x,y) = \cos(2x) - \cos(2y)$

$f(x,y) = \ln\left(4(x^2+y^2)\right)-2$