证明:f(x,y)=|xy|在点(0,0)处连续,fx(0,0)与fy(0,0)存在,在(0,0)处不可微.

问题描述:

证明:f(x,y)=

|xy|
在点(0,0)处连续,fx(0,0)与fy(0,0)存在,在(0,0)处不可微.

证明:∵

lim
(x,y)→(0,0)
f(x,y)=0=f(0,0)
f(x,y)=
|xy|
在(0,0)连续
fx(0,0)=
lim
x→0
f(x,0)-f(0,0)
x
=0
fy(0,0)=
lim
y→0
f(0,y)-f(0,0)
y
=0

f(x,y)=
|xy|
在(0,0)的两个一阶偏导数存在.
∵△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|
(△x)2+(△y)2

若令△x=rcosθ,△y=rsinθ,则有
△f(0,0)
ρ
=
r2|sinθ•cosθ|
r
=r|sinθ•cosθ|
lim
ρ→0
△f(0,0)
ρ
=
lim
r→0
r|sinθ•cosθ|=0

f(x,y)=
|xy|
在(0,0)处可微.