I'm analysing this function from my calculus class:
$$f(x,y) = x \cdot y \cdot \sin(1/x)\cdot \sin(1/y)$$
I want to know if it's possible to redefine this function to all of $\Bbb R^2$ so it's continuous.
Will it suffice to make it equal to 0 along both axis?
Using the fact that $$\forall X\neq 0 \;\;\;|X\sin (\frac 1X)|\leq |X|$$ we get $$\lim_0 X\sin (\frac 1X)=0$$
thus, to be continuous at $\mathbb R^2$, Put $f (x,y)=0$ if $xy=0$ and this is what you said.