I have the following function
$$f(x,y) = \begin{cases} \frac{x \sqrt{|y|} }{2x^2+|y|} &(x,y) \neq (0,0) \\0&(x,y) = (0,0)\end{cases}$$
Is this function continuous at $R^2$?
I've tried to prove that it is not continuous at $(0,0)$, but did not succeed.
Would much appreciate help.
Thank you!
Best Answer
For a function to be continuous at $(x_0,y_0)$ in $\mathbb{R}^2$ it is not enough that the partial limits in the axes $x=0$ and $y=0$ exist and be equal, as I'm supposing you tried (it's a common mistake). You need to guarantee that this limit is the same for all paths going through the point.
For instance, in this case, the axes limits are both equal: $$ \lim\limits_{y\to0}f(0,y)=\lim\limits_{x\to0}f(x,0)=0. $$ But you can also take the parabola $y=x^2$ and check that $$ \lim\limits_{x\to0^\pm}f(x,x^2)=\lim\limits_{x->0^\pm}\frac{x|x|}{3x^2}=\pm\frac{1}{3}. $$ Since they are different, the limit doesn't exist.
A little visualization goes a long way, I recommend Mathematica if you have access to a license.