Continuity of a two variable function

**mrmaaza123** · November 14th 2012, 02:33 AM

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 .

**HallsofIvy** · November 14th 2012, 09:01 AM

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

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

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