real analysis - Why "a function continuous at only one point" is not an oxymoron?

I understand that there are functions that, by definition of continuity, can be continuous at only one point, such as

$$f(x)=\begin{cases}
x,&\text{if }x\in\Bbb Q\\
0,&\text{if }x\in\Bbb R\setminus\Bbb Q\;.
\end{cases}$$

which is continuous only at $x=0$.
But it is continuous because it satisfies the formal definition of continuity. Still,
continuity at only-one-point sounds like an oxymoron to my mind. I understand that mathematical concepts are different than the standard meanings of the words in natural languages, so my question is this:

Does the classical definition of continuity fail to capture the intended concept of continuity for this pathological case? Has anybody attempted to modify the definition of continuity to make this pathological cases fail? I call it pathological because I imagine that, historically, the original concept of continuity attempted to capture the idea of "connected" line. But I might be wrong.

Answer

Via nonstandard analysis, there is a "infinitesimal segment" approach to continuity-at-a-point: a function $f$ is continuous at a point $a$ if, for any infinitesimal $h$, the difference $f(a+h)-f(a)$ (actually, ${}^*f(a+h)-{}^*f(a)$, to be precise; see below) is infinitesimal. That is, $f$ is continuous at $a$ if the graph of $f$ "infinitesmially near" $a$ varies only infinitesimally from a straight (horizontal, in fact) line. This definition is equivalent to the usual $\epsilon-\delta$ definition.

---

OK, so what is this "nonstandard analysis" I mention? Well, this is certainly too complicated for a short paragraph, but basically the idea is this: we start with the "real" universe of the real numbers $\mathbb{R}$ and all functions (continuous or not) on $\mathbb{R}$, and we consider a "blown up" version of the reals, called $^*\mathbb{R}$. $^*\mathbb{R}$ is an ordered field which contains the real numbers, and a lot more junk besides, including field elements which are less than every (real) positive real number; we call these infinitesimals. Moreover, each function $f$ on $\mathbb{R}$ has a version $^*f$ defined on $^*\mathbb{R}$, which agrees with $f$ on the actual reals. Statements in the $\epsilon-\delta$ language can be translated to (arguably) more intuitive and snappy definitions in terms of infinitesimals.

If this all seems suspicious to you, that's very reasonable; it takes a lot of work to set this up so it doesn't break. There are many good sources with actual details; I like http://homepages.math.uic.edu/~isaac/NSA%20notes.pdf, but I'm biased (I learned from it). See especially section 2, which explains why we can get away with this sort of silliness :D.