Doubt on understanding continuity . Just preparing for my multivariable-calculus exam and wanted to clear these things:  
I've come across many questions of sort below ,especially 2-dimensional regions, and wanted to understand the Idea behind them....
Prove the continuity of $f(x,y)$ on $\mathbb R^2$where, 
$$f(x,y) = \begin{cases} \text{some fn./value is given} & \text{, if x,y in region1 } \\ \text{some other fn./value is given} & \text{, if x,y in region2} \end{cases}$$  
Here ,region $1$ and region $2$ consist of all those points $(x,y)$ satisfying respective inequalities in $x$ and $y$...   

to clearly understand my above statements consider example:  $$f(x,y)=
\begin{cases}  e^{-\text(\frac{1}{|x-y|})} & \text{if $x\neq y$} \\ 0 & \text{if $x=y$} \end{cases} $$  

Now if I've to prove continuity on $\mathbb R^2$ :  
STEP 1: I should pick up any $(x_0,y_0)$ in $\mathbb R^2$ where continuity can be proved, 
STEP 2: Now what I've to show that limit of $f(x,y)$ where $(x,y)$ are in region $1$ must be equal to limit of $f$ at $(x_0,y_0)$.
similarly ,show the above for region $2$.  
Am I correct with this procedure..... 

Step 1: prove that the function is continuous whenever at a point $(x_0,y_0)$ whenever $x_0\ne y_0$. Should be evident since the function is a composition of continuous functions.
Step 2: prove that the function is also continuous at the points $(x_0,x_0)$ for arbitrary $x_0$. Should be also doable. Hint: $$|f(x,y)-f(x_0,x_0)| =|f(x,y)-f(x_0,y)+f(x_0,y)-f(x_0,x_0)|. $$
