选 B）.事实上,由于
lim((x,y)→(0,0))[f(x,y)/(x^2 + y^2)]
存在,可知应有 f(0,0) = 0.于是
f'x(0,0) = lim(x→0)[f(x,0) - f(0,0)]/x
= lim(x→0){[f(x,0) - f(0,0)]/x^2}x = 0,
同理,f'y(0,0) = 0；进而,由
lim((Δx,Δy)→(0,0))[f(Δx,Δy) - f(0,0) -f'x(0,0)Δx - f'y(0,0)Δy]/[√(Δx^2 + Δy^2)]
= lim((Δx,Δy)→(0,0))[f(Δx,Δy)/(Δx^2 + Δy^2)][√(Δx^2 + Δy^2)] = 0,
得知 f(x,y) 在(0,0) 处可微.