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Sorry for what must be a very simple question, but Internet searches have failed me.

Is there a standard way to represent "continuous on some range"?

For instance, if I want to say

$g^{\prime}$ is continuous on $[a,b]$

, is there a way to represent the English words "is continouus on" by some standard symbol? I'm not looking for a definition of continuity, merely a standard symbol that represents the idea typographically.


[Edit, 2011.05.27:]

I've been asked to be more precise about the function. Actually, this example was taken from the first part of a published statement of the Substitution Rule for definite integrals, which I have been trying to put tersely on a flashcard using TeX. The full entry (James Stewart, Essential Calculus [N.p: Thomson, 2007] p.239 [Sec. 4.5]) reads:

If $g^{\prime}$ is continuous on $[a,b]$ and $f$ is continuous on the range of $u=g(x)$, then $$\int_a^bf\left(g(x)\right)g^{\prime}(x)~dx=\int_{g(a)}^{g(b)}f(u)~du$$

It's quite true that "range" appears in reference not to the interval $[a,b]$ but rather to $g(x)$. So I should further ask: is there a way to represent "continuous on the range" that is different from the representation "continuous on the interval", described by one of the commenters?

Really, as a philologist, I am boundlessly impressed at the typographical creativity of mathematicians and was sorry not to find a ready-made symbol anywhere for "continuous". Though I see that the notion of continuity is closely connected to notions of interval and boundedness.

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    that would be $g'\in C[a,b]$. C[a,b] is the set of all continuous functions whose domain is [a,b]2011-05-27
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    The word *range* is usually where the function is "going to", while $[a,b]$ in your case is the *domain* is where the function is "coming from".2011-05-27
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    This is perhaps an unhelpful comment, but: your boxed statement fits neatly in a rather small box, so it would certainly fit on a standard-sized index card. Using too many abbreviations -- especially nonstandard ones -- can make things harder to read. I can't think of any clearer, more concise way to say "continuous on the range of $g$" than "continuous on the range of $g$": more symbols would certainly not help me here.2011-05-27
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    @Pete: Actually, as a matter of learning style, I find conditions easiest to retain if they are expressed very tersely around the periphery. It is the yolk of the flashcard that is hardest to swallow if it is boiled too hard. My cards are a little small — I am using the `avery5371` option of LaTeX's wonderful `flashcards` document class. Thanks, in any case, for your comment.2011-05-27
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    @texmad: by inserting the word "me" in my last comment, I hoped to convey that I was expressing a personal preference. The issue of exactly when the use of symbols makes mathematics easier to read and when it makes it harder is a complex one (and possibly of philological/linguistic interest, I would suggest). As someone who has been reading and writing latexed mathematics for a good while now, I have at least learned what I like and try to write in a way which pleases myself, at least. For me, what is written in the box above is essentially optimal.2011-05-27
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    As someone who is (like most mathematicians) also a teacher of mathematics, let me also say: all but the most precocious math students in the first half of their undergraduate career tend to use more symbols and fewer words (and especially, complete sentences) than I and most of my colleagues would like. A theorem *is* a sentence -- this is something that the logicians worked out formally, but it is equally true in the informal sense -- and taking out too many words runs the risk of interfering with the grammar of the sentence and thus its meaning.2011-05-27
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    (continued) One of the more distressing things to happen as an instructor is to ask (on an exam, say) for a definition or a theorem and get as an answer something which isn't even a well-formed sentence. I definitely call attention to grammatical and spelling mistakes on exams (and homework, when I grade it myself!) -- I don't take points off for it, but only because I think that would be badly received. In summary: mathematicians like words, perhaps more so than in many other technically-minded fields. (For instance, mathematicians use rather few acroynyms, especially compared to CS(!).)2011-05-27
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    +1 to all of Pete's comments above. I personally accepted the use of English words in mathematics very quickly, although it was only in the past year or so that I started to use proper punctuation in displayed equations.2011-05-27
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    Another +1 to all of Pete's comments. There seem to be a few stages that most students of mathematics go through: (i) not using enough symbols because they don't understand why they are necessary, and hence being imprecise (ii) using too many symbols, because they confuse being precise with being formal (iii) recognising when an English sentence conveys meaning more clearly than a string of symbols, and mixing English with symbols in a way which conveys meaning most effectively. I'm not sure I'm at (iii) yet, though it's something I always aim for.2011-05-27

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I would propose $g'|_{[a,b]}\in C([a,b])$, because it sounds like you might want the possibility of the function $g'$ being defined on a larger set to remain open. In that case, we should be precise; we need to restrict $g'$ to a function whose domain is in fact $[a,b]$ before claiming it is an element of $$C([a,b])=\{f:[a,b]\rightarrow\mathbb{R}\mid f\text{ is continuous}\}.$$

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    It is striking to me that the one thing I had hoped to replace in the original expression, the English word "continuous", is still found in your equation!2011-05-27
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    @texmad: No, I proposed the expression $$g'|_{[a,b]}\in C([a,b])$$ Below that, I was merely pointing out the definition of $C([a,b])$ for the purpose of explaining why $g'|_{[a,b]}\in C([a,b])$ makes sense and $g'\in C([a,b])$ doesn't (necessarily) make sense.2011-05-27
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    I have to say that I think you are being overly cautious here. Notation is good if the reader has only one reasonable guess as to what it should mean. To me $f \in C([a,b])$ could mean only one thing. For instance, if $f: \mathbb{R} \rightarrow \mathbb{R}$ is the function which is $1$ at $x = 0$ and $0$ elsewhere, I would have no problem with ''$f \in C[1,10]$'' -- of course this means the restriction of $f$ to $[1,10]$: what else?2011-05-27
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    @Pete: I would probably say $f\in C[1,10]$ in real life, if nothing else to avoid an endless stream of $|\,$'s, but I would say it with a tiny bit of hesitation. I was already kind of "on alert" to the issue because the domain of $g'$ was not specified in the question; but furthermore, I felt that this kind of identification of $g'$ with its restriction to a subset was a "post-rigorous" conception of function, to use [Terry Tao's terminology](http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/),2011-05-27
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    while the nature of the OP's question indicated to me that they are likely thinking in a "rigorous" mindset, so I wanted to give an answer suited to them. In fact I would have to admit that I haven't studied real analysis enough to confidently speak in a post-rigorous manner about it, which perhaps explains my instinct to post the answer that I did. In short: you're right, this is an overly cautious answer.2011-05-27
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If you're really just looking to save space, I will suggest my abbreviation: Simply write "cts" instead of "continuous." "Continuous" is a long word that, in my experience, is difficult to type. "Cts" is very compact and takes away no meaning from the statement--it even preserves the English/symbols hybrid approach that many people have promoted here (and I agree with).

I even defined a macro "\newcommand{\cts}{continuous}" in my TeX documents so that my code is also consistent and easy to type.

This is a matter of choice, so if you don't like it, don't use it. Just my two cents.