# Thread: Redefining a function in R2 to be continuous

1. ## Redefining a function in R2 to be continuous

Let f(x,y) = sin(xy)/x, for $\displaystyle x \neq 0$.
How would you define f(0,y) for y in R so that f is a continuous function
on all of R2

my answer: (*note the back of the book said use f(0,y) = y)
We have to define f(0,y) so that f(0,y) $\displaystyle \rightarrow$ 0 as (x,y) $\displaystyle \rightarrow (0,0)$

By a theorem (stated in the book) we already know f(x,y) is a continuous at all points except where x = 0.

So let f(0,y) = $\displaystyle y^{2}$
$\displaystyle y^{2} \rightarrow 0$ as (x,y) $\displaystyle \rightarrow 0$

professor's comments: Also trouble at ALL points (0,y). Not just (0,0)

I dont really see what my professor is talking about...Every single "path" to the orgin must lead to the same result. Moving along the x axis leads us to 0, so moving along the y axis must lead to 0. $\displaystyle y^{2}$ is clearly a continuous function...and I do not see the difference between the back of the book's answer f(0,y) = y and my answer...

Please show me what I did wrong. Much appreciated.

2. Here's a thought - but note that I am not as smart as you!

I think he is trying to say that there is "trouble" at all points of the function f(0,y), not just the point where y=0. So, I think you should have shown that the function is continuous at all points (0,y), and not just the point (0,0).

Here is what you did - you showed that the function was continuous for all points (x,y) where x is not 0, and then you showed that the function was continuous at the origin. This leaves many points of the form (0,y) where y is not 0.

For example, is f continuous at (0,2)?