Continuity of a piecewise function of two variable I'm given this equation:
$$
u(x,y) = 
\begin{cases}
\dfrac{(x^3 - 3xy^2)}{(x^2 + y^2)}\quad& \text{if}\quad (x,y)\neq(0,0)\\
 0\quad& \text{if} \quad (x,y)=(0,0).
\end{cases}
$$
It seems like L'hopitals rule has been used but I'm confused because


*

*there is no limit here it's just straight up $x$ and $y$ equals zero.

*if I have to invoke limit here to use Lhopitals rule, there are two variables $x$ and $y$. How do I take limit on both of them?


Probably, it is a piecewise definition.  $u(x,y)$ makes sense as long as both $x$ and $y$ are not $0$.  It is probably just to mean that: if $(x,y)\neq (0,0)$ then $u(x,y)$ is as defined (with the formula), otherwise $u(x,y)=0$.  You certainly cannot plug in $(0,0)$ to the formula.  It may be that the point is chosen because there is a "hole" and that $0$ would fill in the hole, but this is not necessarily the case.
