Please verify that this function is continuous everywhere on its domain In high school, I was taught the "naive" definition of continuity of a function. Consider
f(x) = x, where the domain of f = {x: 0 ≥ x ≥ 1} ∪ {x: 2 ≥ x ≥ 3}.
Back in high school, I would have thought that because there's a break in the graph, that f(x) is discontinuous. Now, after reading the definition of a continuous function in my real analysis book, I am convinced that this function is continuous everywhere on its domain. Am I correct?
 A: You're right, this function is continuous, as you can easily verify using the definition of continuity in an analysis book.
But what about the "high school definition" of continuity? Of course it's not really a definition, but let's accept it for the moment as a kind of useful "intuitive definition" for pedagogical purposes. 
That definition could easily be reformulated in a more useful and more general manner which applies to domains like the one in your post, something like this:

Over any portion of the domain which has no break, there's also no break in the graph. 

If one has a domain, like the one in your post, which is a union of some number $n$ of intervals with $n-1$ breaks in between them, one can be more precise, like this: 

Over any interval contained in the domain, there's no break in the graph.

Applying either one of these "definitions" (and using the correction suggested in the comment of @fleablood), over the first portion $\{x : 0 \le x \le 1\}$ of the domain the graph of $y=x$ has no break; then there's a break in the domain; and then, over the second portion $\{x | 2 \le x \le 3\}$ of the domain, the graph of $y=x$ also has no break. So the function is continuous.
Eventually, though, the "high school definition" is going to break down in an irretrievable manner, particularly for domains which are not just finite unions of intervals. Perhaps the simplest such domain is the set $\{0\} \cup \{\frac{1}{n} \mid n \in \mathbb N\}$. The analysis definition can tell us whether or not a function with this domain is continuous, but the "high school definition" would be useless.
