I would like to study the continuity on $\mathbb{R}^2$ of these functions :
$ 1) f(x,y)=\frac{y}{x^2}e^{-\frac{|y|}{x^2}}$ if $x \neq 0$
$0$ if $x=0$
$ 2) f(x,y)=(x^2+y^2)\sin(\frac{1}{xy})$ if $xy \neq 0$
$ 0$ if $xy=0$
Which is the common way to day it with functions of two variables ?
Thank you
On
For 2., note $xy = 0$ iff $(x,y)$ lies on the $x$-axis or on the $y$-axis. Off the axes, $f(x,y)$ is $C^\infty,$ hence is continuous in those regions. Consider a point $(a,0), a\ne 0.$ Then $f(a,y) = (a^2 +y^2)\sin (1/ay).$ As $y\to 0,a^2 +y^2\to a^2>0,$ while $\sin (1/ay)$ oscillates crazily between $-1$ and $1.$ Thus $f$ is not continuous at $(a,0).$ The same thing happens at any $(0,a),a\ne 0.$ At the origin, however, $|f(x,y)| \le x^2 + y^2,$ so $f$ is continuous at $(0,0).$ (In fact $f$ is differentiable at $(0,0).$)
The only point to be studied for continuity is the origin.
1) Take $y=x^2$ you have $$f(x,y)=\frac{y}{x^2}e^{-\frac{|y|}{x^2}}=e^{-1}$$ Hence $f$ cannot be continuous as $f(0,y)=0$
2) You have for all $(x,y) \in \mathbb R^2$: $$\vert f(x,y) \vert =\left\vert (x^2+y^2)\sin(\frac{1}{xy})\right\vert \le x^2+y^2$$ Which proves that $\lim\limits_{(x,y) \to (0,0)} f(x,y)=0$.
How to find that?
Look at the function along paths like $y=kx$ or $y=\vert x \vert^p$. If the limit exists along those paths, try to prove that the limit exist... until you get a contradiction or a valide proof. If the limit doesn't exist, you're done.