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I just realized something that was left unnoticed by me for many years. Apparently, among German speakers reelle Funktion (literary also translated word by word as "real functions") has both domain and codomain as real or subsets of reals (most German textbook for math undergraduate defines it this way). While in english, when we say real function, we really mean a function that takes real values (and no other extra conditions). So in english only the codomain has to be a subset of the real numbers.

I just want to confirm the general notion of reelle Funktion among German mathematicians. I don't know how German researchers call real functions the way we know it (as in real valued functions). Is reelle Funktion polysemic (i.e. has multiple meanings) in German?

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    $\begingroup$ "Reelle Funktion" means, that both, domain and codomain are subsets of the reals. If only the codomain is subset of the reals, we say "reellwertige Funktion". $\endgroup$ – Tim B. May 15 '15 at 17:54
  • $\begingroup$ Sorry, can't help you with your follow up question. As I'm german, I don't know about the english terminology that well. But glad that i could help you with the first one :) $\endgroup$ – Tim B. May 15 '15 at 18:13
  • $\begingroup$ Oops got my last comment deleted.. apparently in english there is no shorter way than saying "real function with real domain" or "real valued function of a real variable" $\endgroup$ – Jose May 15 '15 at 18:15
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I am german, and I would call a function, regardless of its domain of definition, which takes real values 'eine reellwertige Funktion' (which is pretty much the literal translation from the english, taking into account the fact that the german language often concatenates two or more words, hence the transition from 'real valued' to 'reellwertig'). Similarly, there is 'eine komplexwertige Funktion' for a function which takes complex values, but whose domain is not necessarily a subset of the complex numbers.

From my experience, 'eine reelle Funktion' is really, as you say, a function whose domain and codomain are both the real numbers (or possibly subsets thereof).

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Doing a quick search with google reveals, that most people use "reelle Funktion" for a function $f : \mathbb{R} \rightarrow \mathbb{R}$, others use it for a function $f : D \rightarrow C$ with $D,C \subseteq \mathbb{R}$, where $D$ or $C$ may or may not be open or closed intervals and some use it for $f : D \rightarrow \mathbb{R}$, where $D$ can be any set. In any case, you should check, how the author defines the term and personally I would avoid using it generally and instead speak of "einer Funktion von R in das offene Interval a,b" or "einer Funktion von einer Teilmenge D der reellen Zahlen in die reellen Zahlen" or something similar, and "Zahlenfolge", if the function is a sequence of reals. This is also, what I'm familiar with.

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