Delete this SOP.—msh210℠ (talk) 17:24, 19 October 2010 (UTC)
Keep outside CFI, as a set phrase; it looks like a set phrase, and has 165,000 Google Books hits. This looks sum-of-partish, but it is a construction peculiar to English as compared to Czech: there is "reálná funkce" in Czech but there is neither "reálně-hodnotová funkce" not "reálně-hodnocená funkce" in Czech. If you delete this, it should better be documented somewhere that "real-valued function" is a common synonym of "real function", possibly in "real function" entry. The entry "real function" can be kept so long until someone adds to the entry "real" the definition "Of a function, such that its codomain is a subset of reals" :p. After both entries get deleted, let us think of the best place in which the commonality of the terms "real function" and "real-valued function" can be documented. --Dan Polansky 07:20, 20 October 2010 (UTC)
But it's not even a set phrase. Consider all the hits for "function is real-valued".—msh210℠ (talk) 19:35, 20 October 2010 (UTC)
And "number is prime" — but prime number unfortunately passed. Equinox◑ 23:34, 20 October 2010 (UTC)
You have used the same argument about separability in "prime number" and "free variable", and the same argument applies to the list of items that I have given in the paragraph below, including "continuous function". The argument relies on the exclusion principle to the effect of "If a two-word term consisting of a noun and an adjective is separable, then it should be excluded". If this exclusion principle is accepted, most of the items given the list below get excluded, including "continuous function" and "free software". I reject this exclusion principle. --Dan Polansky 06:08, 21 October 2010 (UTC)
This can be argued even more broadly, by proposing this principle: "Some separable multi-word terms should be included". Terms that come under this principle include separable phrasal verbs such as "look up" ("look it up"). --Dan Polansky 06:32, 21 October 2010 (UTC)
Actually, this seems to come within purview of the arguments in Talk:prime number and Talk:free variable. The trick how to render sum-of-partish a term of the form "A B", one defined as "B such that R", is to add to "A" the definition of the form "Of B, such that R". Here, A is "real-valued, and B is "function". Some of such entries: algebraic number, algebraic integer, bound variable, cardinal number, complex number, free variable, imaginary number, rational number, real number, transcendental number, free software, open set, closed set, complete graph, normal distribution. --Dan Polansky 07:41, 20 October 2010 (UTC)
AFAICT, this collocation fails the modification and coordination tests for a set phrase.
It accepts intervening modifiers such as "Liapunov", "locally Lipschitz", "measurable", "harmonic"
"Real-valued" coordinates via "and" (sometimes also "or") with terms such as "complex-valued", "bounded below", "non-decreasing", "continuous"
"Function" coordinates with "mapping" and "variable".
Further, I reject the DanP rejection principle in favor of the CGEL set-phrase criteria just applied. This is not a case of a few rare instances of the collocation failing the test. It seems absolutely transparent that "real-valued" is a simple attribute of a "function". DCDuringTALK 23:57, 21 October 2010 (UTC)
Real-valued does say "of a function". If we can demonstrate that it collocates with other nouns, then this must be SoP. Mglovesfun (talk) 09:59, 22 October 2010 (UTC)
There are hits for "real-valued parameter", "... random variable", "... operator", "... functional" (noun), "... martingale", "... random walk", and "... Brownian motion", for example. (There are also hits for "real-valued vector", but that's probably (I'd have to think about it and check citations) for a different sense of real-valued.)—msh210℠ (talk) 16:42, 22 October 2010 (UTC)
Delete since it belongs at real-valued, although like Dan I do wish there were a way to preserve common collocations. DAVilla 10:26, 27 October 2010 (UTC)