What does "$f: [a,b] \to \mathbb R$ on a compact interval" means?

Question

In some theorems, I see "$f: [a,b] \to \mathbb R$ on a compact interval". $[a,b]$ is actually compact, does this kind of emphasize have another meaning?

Edit for a comment: "A continuous function $f: [a,b] \to \mathbb R$ on a compact interval is Riemann integrable."

Could you quote some more context for that phrasing, please? — hmakholm left over Monica, Nov 13 '16 at 19:40
The edited comment still looks like it's a frament . The property is not true for all $f$ -- for example, the indicator function of $\mathbb Q\cap [a,b]$ is not Riemann integrable. — hmakholm left over Monica, Nov 13 '16 at 19:44
Ah, sorry, I forgot to put it is continuous. By the way, what does frament mean? I couldn't find it in dictionary. (I am learning English too:) ) — student forever, Nov 13 '16 at 19:47
I think just emphasis. Or giving it a name. Say "compact" interval instead of "closed" interval because, technically, $[a,+\infty)$ is a "closed" set and you do not want to include this. — GEdgar, Nov 13 '16 at 20:14

score 1 · Accepted Answer · answered Nov 13 '16 at 20:09

It's mostly just redundant emphasis: the phrase "on a compact interval" and writing the domain as $[a,b]$ are basically conveying the same information twice.

However, the phrase "on a compact interval" also clarifies that $[a,b]$ can be any compact interval and the claimed statement will hold. If you leave it out, it might sound like you are referring to some specific interval $[a,b]$, where you have previously defined $a$ and $b$, and that it might not hold for more general compact intervals.

As I understood by comments too, there is no further meaning. Thanks for your answer. — student forever, Nov 13 '16 at 20:18

What does "$f: [a,b] \to \mathbb R$ on a compact interval" means?

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