If a function is not defined at a point, is it also not continuous there?

Question

Suppose we adopt the following definition of continuity:

DEFINITION: Let $f$ be a function defined in a neighbourhood of a given point $a$. We say that $f$ is continuous at $a$ if $$ \lim_{x \to a}f(x)=f(a) \, . $$

Consider the function $f(x)=1/x$. Since the domain of $f$ is $\mathbb{R}\setminus\{0\}$, the question 'is $f$ continuous at $0$?' seems non-sensical. The definition of continuity only applies when the function is actually defined at the point under consideration. Because of this, I have seen many contributors on MSE write 'the function $f$ is neither continuous nor discontinuous at $0$'. However, I feel uncomfortable making such a statement. To me, the question itself doesn't make any sense, and so it's not even possible to say that the function is neither continuous nor discontinuous. Rather, nothing can made of the question. So does it make sense to say '$f$ is neither continuous nor discontinuous at $0$?'

I'm not very familiar with logic, but I wonder whether it makes sense to argue that the statement '$f$ is continuous at $0$' does not have a truth value, as it is non-sensical.

Answer 1

The answer is: It depends on how everything is defined. There are slight differences in how one sets up the definitions (nothing substantial), but these differences can make the statements you're asking about a bit more complicated.

In your definition for continuity, the first line states that $f$ must be defined in a neighborhood of $a$. In your example, $f$ is not defined in a neighborhood of $0$, so the first condition in the definition fails and the definition doesn't apply.

The way that this version of the definition is phrased, it doesn't discuss continuity when $f$ is not defined in a neighborhood of $a$ (i.e., either $a$ is not in the domain or $a$ is an isolated point of the domain). Conceivably, one could introduce definitions for continuity in these cases without introducing a contradiction, but continuity is such an established notion that this would be confusing. Without such an additional definition, if $f$ isn't defined in a neighborhood of $a$, then it cannot be continuous.

For instance, it is very common to extend the definition of continuity for endpoints of an interval (such points would not have a neighborhood in the domain) and say that a function is continuous at an endpoint of its domain if it has one-sided continuity there.

For discontinuity, it depends on how this is defined. Perhaps discontinuity is defined as the complement of continuity. Then, a function is continuous or discontinuous at every point.

Perhaps discontinuity is only defined for points in the domain. In this case, points which are not in the domain are neither continuous or discontinuous.

Discontinuity could be defined for points in the closure of the domain. In such a case, there are points not in the domain for which the function is discontinuous.

Choosing each of these definitions leads to slightly different notions of discontinuity (and one should be clear about which one you're discussing). In most cases, choosing one definition over another isn't significant, but leads to differences in what descriptors you can use for various points.