Partial derivatives do not guarantee continuity

Even if a function z = f(x,y) has partial derivatives, it need not be continuous!

For example, consider f(x,y)=(((x^3)-xy)/((x^2)+(y^2)))for (x,y)≠(0,0) and f(0,0)=0.

Then f(x,0)=x and f(0,y)=0, so that (Subscript[f, x])(0,0)=1 and (Subscript[f, y])(0,0)=0. (See red lines below.)

However, f(x,x) ⟶ -1/2 ≠ 0 when x ⟶ 0, so that f(x,y) is not continuous at (0,0). (See green line below.)

f[x_,y_]=(x^3-x*y)/(x^2+y^2);
xDirection=ParametricPlot3D[{x,0,f[x,0]},{x,-5,5},PlotStyle->{Red,Thick}];
yDirection=ParametricPlot3D[{0,y,f[0,y]},{y,-5,5},PlotStyle->{Red,Thick}];
OtherDirection=ParametricPlot3D[{x,x,f[x,x]},{x,-5,5},PlotStyle->{Green,Thick}];
xyGraph=Plot3D[(x^3-x*y)/(x^2+y^2),{x,-5,5},{y,-5,5},PlotPoints->50,AxesLabel->{x,y}];
Show[xyGraph,xDirection,yDirection,OtherDirection]