# Thread: Continuity of a two variable function

1. ## Continuity of a two variable function

The question is :

Prove that f is continuous at the point (0,0) where f is

f(x,y)= ysin(1/x) , when x is not equal to zero
0 , when x=0.

I have to prove this using the epsilon-delta definition of limit. Can't figure out how to do that ?

Can somebody please explain the procedure .

2. ## Re: Continuity of a two variable function

Do you not know what the "epsilon-delta definition of a limit? For a function of two variables, you must show
"For any $\displaystyle \epsilon> 0$ there exist $\displaystyle \delta> 0$ such that if $\displaystyle \sqrt{x^2+ y^2}< \delta$ then $\displaystyle |f(x, y)- L$< \epsilon[/tex].

Here, your function is $\displaystyle y sin(1/x)$ and you want the limit to be the value of the function, 0. So you want to get $\displaystyle |y sin(1/x)|< \epsilon$ for x and y sufficiently small. I would use the fact that $\displaystyle |y sin(1/x)|= |y||sin(x)|\le |y|$ because $\displaystyle -1\le sin(1/x)\le 1$ for any x not equal to 0.