Question

第六题 证明f(x,y)在点（0,0）处可微,急.

Answer 1

不妨将条件写为(x,y) → (0,0)时, (f(x,y)-f(0,0)-2x+y²)/(x²+y²) = -1/2+o(1).

于是f(x,y) = f(0,0)+2x-y²-(x²+y²)/2+o(x²+y²) = f(0,0)+2x+o(√(x²+y²)).

其中用到(x,y) → (0,0)时0 ≤ y²/√(x²+y²) ≤ √(x²+y²) → 0.

而等式f(x,y) = f(0,0)+2x+o(√(x²+y²))即说明f(x,y)在(0,0)可微.