Let M(t) be defined on R:C, where R and C is the set of real and complex numbers respectively. Suppose T is the smallest real number, t for which M(t) is real. Given the existence of such a T, the only solace I got is that it was fairly small.
A function of a real variable is a function whose domain is an improper subset of the real line. Functions of a real variable were the classical object of study in analysis, specifically real analysis. In that context, a function of a real variable usually meant a real-valued function of a real variable, that is, a function whose domain and codomain were the real numbers.
However, as real analysis evolved, because of their convenience in fields such as Limits, 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.