Multiply periodic real functions and their period sets

Multiply periodic real functions are rather pathological; continuous ones have to be constant and, as will be shown, measurable ones must be constant a.e. Beyond this, some other properties of these functions are discussed, and examples are given. Since the set of periods of a real function is a subgroup of (R,+), it is possible to apply algebraic methods to get a better survey. It will be shown that the homogeneous sets defined by Borel are just cosets of these subgroups; this leads to a transfer of theorems about homogeneous sets to results about period sets (and, at the same time, subgroups of (R,+)).