A function for which both the set of definition and the set of values are subsets of the set of real numbers.
So, a real function is understood to be a real-valued function on a subset of the real numbers. In the Western literature, "real" most often simply means "real-valued" .
Real function. L.D. Kudryavtsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Real_function&oldid=17448