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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$\begingroup$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 $$C([a,b])=\{f:[a,b]\rightarrow\mathbb{R}\mid f\text{ is continuous}\}.$$
$\endgroup$ 5 $\begingroup$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.
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