举一个某个二元函数的两个混合二阶偏导数在某一点的值相等但在该点这两个混合二阶偏导数不连续的列子.

Question

举一个某个二元函数的两个混合二阶偏导数在某一点的值相等但在该点这两个混合二阶偏导数不连续的列子.

Answer 1

F(x,y)=x^3y^3sin(1/(xy)),xy≠0.
F(x,y)=0,xy=0.
1.xy=0,显然有
Fx'(x,y)=Fy'(x,y)=0.
2.xy≠0,
Fx'(x,y)=3x^2y^3sin(1/(xy))-xy^2cos(1/(xy)),
Fy'(x,y)=3x^3y^2sin(1/(xy))-x^2ycos(1/(xy)).
3.
xy=0,显然有
Fxy''(x,y)=Fyx''(x,y)=0.
4.
xy≠0,
Fxy''(x,y)=Fyx''(x,y)=
=9x^2y^2sin(1/(xy))-5xycos(1/(xy))-sin(1/(xy)).
==>
在R^2上,F(x,y)的二阶混合偏导数相等,
但是二阶混合偏导数不连续.