Continuity is not the necessary condition for partial derivative to exists

Question

I come accros the result the a function of two variable may have partial derivatives of first order at a point though it is not continuous there for example f(x,y)=xy/x²+y²,(x,y)≠(0,0) and f(0,0)=0 is not continuous but have partial dervatives fx and fy=0 at (0,0).Now a natural question arises partial derivatives are slope to tangent at the curve formed by cutting surface with plane y=y0 then if surface is not continous at origin then then any curve formed by cutting plane about y=0 will not be continuous at origin .Then how the partial derivative exists ?? Please help me to solve out this problem

Answer 1

If you draw a picture of the graph of this function you will see that it's not a "surface" in the sense you imagine. It does behave nicely on the intersections with the vertical planes over the coordinate axes, so it has partial derivatives.

Answer 2

Partial derivatives "see" only on a small subset. Namely, a straight segment near the point.

Consequence: partial derivatives are a bad generalization of the 1D derivative. See Differentiability in higher dimensions.