I need help finding if the limit of $g(x,y)$ at $(0,0)$ and at $(2,0)$ to see if it is continuous on these points.
I get confused, because normally these piecewise functions are defined such that $(x,y)$ different than some $(a,b)$ but here I have no idea how to do it because I can approach $(0,0)$ or $(2,0)$ from both pieces of the function…
And if I wanted to see the differentiability as well, what should I do?
I would really appreciate if you could explain this to me. Thank you !
Best Answer
For $\vert y \vert \le 1$ and $y \neq 0$, you have
$$\vert g(x,y) \vert \le \vert x y \vert + \vert x \vert \vert y \vert^3 \le 2 \vert x \vert \vert y \vert.$$
As the right hand side of this inequality goes to zero as $y \to 0$ and $g(x,0)=0$ for all $x$, we get that $g$ is continuous everywhere.