analyse
(1) continuity is obvious.
Elementary function, (0,0) is in the defined area.
(2) The existence of partial derivatives.
lim(△x→0)[f(0+△x,0)-f(0,0)]/△x
=lim(△x→0)(0-0)/△x
=0
∴ partial derivative FX (0 0,0) = 0
Similarly, the partial derivative fy (0 0,0) = 0.
(3) Non-differentiable.
△z=f(△x,△y)-f(0,0)=√|△x △y|
fx(0,0)△x+fy(0,0)△y=0
△z-[fx(0,0)△x+fy(0,0)△y]=√|△x △y|
This is not ρ=√(△x? +△y? ) is infinitely small in high order,
Therefore, f(x, y) is nondifferentiable at (0,0).