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?