The following a continuity question and the answer I put. My professor marked me down, I dont understand his comments, please help.

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.