I had the following questions in one of my exercises
Determine if the following function is continuous $$ f(x,y)= \begin{cases} \frac{2xy}{x^2+y^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} $$
The answer to this is No since the limit doesn't exist at $(0,0)$, but to explain the continuity of this function at points other than the origin, my instructor used the polar form function to showcase that the function is well-defined and so is continuous.
The polar form of this function is
$$f(r,\theta) = \sin2\theta $$
He said that since the polar function is well-defined everywhere except for $0$ (perhaps he meant origin I guess) we can say that the function is continuous everywhere except for $0$.
The question I've is
- Why the function is not well-defined at the origin?
- How can we determine continuity at any point using polar form?