Function Partially Differentiable but Not Totally Differentiable

Question

Professor today briefly introduced that there exists a function which partially differentiable but not Totally differentiable.

Which function follows this kind of property? If give, will draw a plot using my software. Thx.

Answer 1

Consider the function $$f(x,y) = \frac{xy}{x^2+y^2}\quad x^2+y^2 \neq 0$$ where $f(0,0) = 0$. This function is defined on $\mathbb{R}^2$ and has partial derivatives everywhere but is not continuous (and therefore not differentiable) at the origin.