These classes of functions were both introduced by Jovan Karamata,[1][2] and have found several important applications, for example in probability theory.
A measurable function L : (0, +∞) → (0, +∞) is called slowly varying (at infinity) if for all a > 0, Definition 2.
[1][2] Regularly varying functions have some important properties:[1] a partial list of them is reported below.
More extensive analyses of the properties characterizing regular variation are presented in the monograph by Bingham, Goldie & Teugels (1987).
This implies that the function g(a) in definition 2 has necessarily to be of the following form where the real number ρ is called the index of regular variation.