Exact definition of continuity of $f$ on $A \subseteq \textrm{dom}_f$

Question

For $f:\mathbb{R} \to \mathbb{R},$ is there a universal definition of "$f$ is continuous on $A \subseteq \mathbb{R}$"?

For example, what about the function $g:\mathbb{R} \to \mathbb{R}$ which takes the value $5$ on $[2,3]$ and the value $0$ on $(-\infty,2)\cup(3,\infty)$?

Clearly $g$ is not continuous at all points in $[2,3]$ as $g$ is discontinuous at $2.$

Clearly $g|_{[2,3]}$ is a continuous function.

Ps. I'm pretty sure I understand stuff logically. Just wondering if there exists an "official" definition? So answers quoting a well-known author such as Rudin or Apostol would be appreciated, thanks [I don't have access to libraries now].

Answer 1

Your function $g$ is indeed not continuous at $2$.

However, the function $h$ defined on the set $[2,3]$ to be the same as $g$ (in other terms, the restriction of $g$ to $[2,3]$), is continuous at $2$. This is not a contradiction, because we're considering two different functions.

Here's a quite common definition of continuity for functions taking on real values:

Let $f$ be defined on a set $A$ of real numbers and let $a\in A$; the function $f$ is said to be continuous at $a$ if, for every $\varepsilon>0$, there exists $\delta>0$ such that, for every $x\in A$ with $|x-a|<\delta$, it holds that $|f(x)-f(a)|<\varepsilon$. The function $f$ is said to be continuous if it is continuous at every point of its domain $A$.

Such a definition ensures that restricting the domain of a continuous function yields a continuous function. Your example shows that the restriction of a non continuous function may be continuous.

Note that specifying the domain of the function we want to analyze the continuity of is essential. On the other hand, textbooks may have a slightly different definition, so you need to check the conventions used and stick with them.