Continuity in the point (0,0)

If the derivative exists at (0,0)

Continuity of the partial derivatives at (0,0)

Directional derivatives at (0,0)

One of the functions is, for example:

$$f(x,y)=\{\begin{array}{ll}\frac{{x}^{2}{y}^{2}}{\sqrt{(}{x}^{2}+{y}^{2})},& \text{if if (x,y) not (0,0)}\\ 0,& \text{if (x,y) = (0,0)}\end{array}$$