# Determining if $f(x,y)$ can be expanded to be continuous on whole $\mathbb{R}^{2}$

I'm trying to determine if the function

$$f(x,y)=\frac{\sin(xy)}{xy+x^3y^3}$$

can be expanded so that it becomes continuous on the whole $$\mathbb{R}^{2}$$-plane. I'm aware of that the function isn't defined for when $$xy= 0$$, so in order to become continuous everywhere the function needs to be expanded to be defined, and continuous when we're located on either axis.

Up to this point I've only worked with problems where the function isn't defined on a single point $$(a,b)$$. When that was the case it was quite simple to solve since I could switch to polar coordinates and expand the function so that $$f(a,b) = \lim_{(x,y)\to (a,b)} f(x,y)$$. However, now when I have multiple points, I can't see how I can solve it the same way.

• This is one of the rare times in multivariable calculus where you could use L'Hopital's rule. The conditions for the straightforward application using directional derivatives numerator and denominator are that both share a continuous set of $0$'s such as a curve. It will not work if the zeros are isolated. Feb 12, 2021 at 22:34

The easiest way for this is studying the limit of $$f$$ on the points where $$xy=0$$, and as for the single-point case, just define the extension on those points as the value of the limit, whenever exists. For studying that limit it might be usefull noting that: $$f(x,y)=\dfrac{\sin(xy)}{xy}\cdot\dfrac{1}{1+x^2y^2}$$ And considering the change of variable $$t=xy$$.
• good grief! Can you not find a more optimistic way of putting that $\ddot{\frown}$? Feb 12, 2021 at 23:23