MATH 3500 3.1: Example Function

This is a 3d plot of a non-differentiable example function whose partial derivatives are all defined.

f[x_,y_]:=Piecewise[ {{(x^2)(y/((x^2)+(y^2))),x^2 + y^2 != 0},{0,True}}]

Plot3D[f[x,y],{x,y} ∈ Disk[{0,0},5]]

We see that at the origin, the partials are well-defined:

Limit[Simplify[((f[0+h,0] - f[0,0])/h)],h->0]

and

Limit[Simplify[((f[0,0+h] - f[0,0])/h)],h->0]

But if we consider the partials as functions of  , they are not continuous at the origin:

Plot3D[Derivative[1,0][f][x,y],{x,y} ∈ Disk[{0,0},5]]

Plot3D[Derivative[0,1][f][x,y],{x,y} ∈ Disk[{0,0},5]]