Real (Valued) Functions in German

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?

• "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". – Tim B. May 15 '15 at 17:54
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.