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).