# Function in two variables - continuity

• May 21st 2009, 09:34 PM
Robb
Function in two variables - continuity
Hello,
Still struggling with these proofs, some help would be appreciated :)
Can the function
$f(x,y) = \frac {\sin x\sin^3 y}{1 - \cos(x^2+y^2)}$
be defined at $(0,0)$ in such a way that it becomes continuous there? Prove your answer.

Regards,
• May 21st 2009, 10:45 PM
NonCommAlg
Quote:

Originally Posted by Robb
Hello,
Still struggling with these proofs, some help would be appreciated :)
Can the function
$f(x,y) = \frac {\sin x\sin^3 y}{1 - \cos(x^2+y^2)}$
be defined at $(0,0)$ in such a way that it becomes continuous there? Prove your answer.

Regards,

no it can't! because $\lim_{(x,y)\to(0,0)} f(x,y)$ doesn't exist: $\lim_{x\to0}f(x,x)=\frac{1}{2}$ but $\lim_{x\to0}f(x,-x)=\frac{-1}{2}.$