Joint and Seperate Continuity Example I am trying to understand joint and separate continuity with the following example: 
$f(x,y) = 0$ if $x\neq 0$ and $y \neq 0$ and 1 if either $x=0$ or $y=0$. 
In each individual variable, $f$ is continuous at $(0,0)$ since the mappings are constant, but why is $f$ itself not continuous at $(0,0)$. Is it because the $lim_{(x,y) \rightarrow (0,0)}=0$ whereas the $lim_{y \rightarrow 0}[lim_{x\rightarrow 0}]$=1?
 A: It is not right to say that $\lim_{(x,y) \to (0,0)} f(x,y)=0$. This limit does not exist. Apparently what you mean, to be said more precisely, is that $\lim_{x \to 0} f(x,x)=0$. Also, $\lim_{x \to 0} f(x,0)=1$. This shows the the limits are different, along "different paths", hence $\lim_{(x,y) \to (0,0)} f(x,y)$ does not exist. One "path" is the line $y=x$, another path is the $x$-axis, and 
along these two paths the function has different limits. These different limits may also be written as $\lim_{(x,x) \to (0,0)} f(x,x)=0$ and $\lim_{(x,0) \to (0,0)} f(x,0)=1$, or also as $\lim_{(x,y) \to (0,0),x=y} f(x,y)=0$ or $\lim_{(x,y) \to (0,0),y=0} f(x,y)=1$. The non-existence of this limit is akin to the case when a function of one variable has different limits from the left and  from the right. 
A: The equation $z=f(x,y)$ describes the plane $z=1$ everywhere except over the x-axis and y-axis, where $z=0$.  Thus if we approach the x,y-origin from any direction along either discontinuity (the axies), then $z=0$, however if we make the approach from any other path, then $z=1$ until we hit the origin.
Hence we have: $$\begin{align}\lim_{x\to 0}f(x,0) & = \lim_{y\to 0} f(0,y) = 0\\ \forall \alpha\neq 1\;\forall \beta\neq 0:\quad \lim_{t\to 0} f(\alpha t, \beta t) & = 1 \\ \therefore \lim_{(x,y)\to(0,0)} f(x,y) & \quad \text{Does Not Exist} \end{align}$$
A multivariate limit must exist along all possible approach paths, not just the obvious or easy ones.
It is oft times much easier to prove such a limit does not exists, by finding the contradictory path, than it is to prove that all paths lead to Romethe same limit.
