Functions of a real variable were the classical object of study in mathematical analysis, specifically real analysis. In that context, a function of a real variable was usually meant a real-valued function of a real variable, that is, a function whose domain and codomain were the real numbers. However, because of their convenience in fields such as Fourier analysis, it was also common to consider complex functions of a real variable, that is, a function whose domain was the real numbers and whose range was the complex numbers.
Until the introduction of functional analysis in the 1920s, the study of functions of a real variable was one of the two major subdivisions of mathematical analysis. The other was the study of functions of a complex variable, otherwise known as complex analysis.