Is g(x,y) a continuous function?

[newyorkcity]

How would I analyze the continuity of:

g(x,y) = sin(2x^2 - y^2) / 2x^2-y^2 unless y^2=2x^2
1 if y^2=2x^2

g(x,y) seems to be continuous for all values of (x,y)... However, I realize that the function assumes the value 0 when y^2=2x^2. I am not really sure how to go further than this... the function seems to be continuous, unless we specify that it assumes 1 when y^2=2x^2, which makes the function discontinuous.

Am I missing something?

 

[Dick]

The function doesn't assume the value 0 if y^2=2x^2, the ratio sin(2x^2 - y^2) / 2x^2-y^2 undefined if y^2=2x^2 because it has a zero denominator. What's the limit of sin(2x^2 - y^2) / 2x^2-y^2 as 2x^2-y^2 approaches zero?