| lim |
| (x，y)→(0，0) |
∴f(x，y)=
| |xy| |
∵fx(0，0)=
| lim |
| x→0 |
| f(x，0)-f(0，0) |
| x |
| lim |
| y→0 |
| f(0，y)-f(0，0) |
| y |
∴f(x，y)=
| |xy| |
∵△f（0，0）=f（△x，△y）-f（0，0）=
| |△x•△y| |
∴△f(0，0)-fx(0，0)△x-fy(0，0)△y=
| |△x•△y| |
∴
| △f(0，0) |
| ρ |
| |△x•△y| | ||
|
若令△x=rcosθ，△y=rsinθ，则有
| △f(0，0) |
| ρ |
| r2|sinθ•cosθ| |
| r |
∴
| lim |
| ρ→0 |
| △f(0，0) |
| ρ |
| lim |
| r→0 |
∴f(x，y)=
| |xy| |