I am wondering how to check whether a multidimensional function is continuous. I.e. I am thinking of functions like
$f_1(x,y)=\frac{x^2y}{x^2+y^2}$ with $f_1(0,0)=0$
$f_2(x,y)=\frac{2xy^3}{(x^2+y^2)^2}$ with $f_2(0,0)=0$
In order to check for continuity at point (0,0) I know to approaches:
1) express x,y in a different coordinate system. I.e. expressing x,y in polar coordinates tells me that $f_1$ is continuous and $f_2$ is not.
2) searching for constant $C$ so that $f < xC$ or $f < yC$
However what can I try if these two criteria are inconclusive? How do you check for continuity of a multidimensional function?
Thanks in advance