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In real analysis, a branch of mathematics, a **slowly varying function** is a function of a real variable whose behaviour at infinity is in some sense similar to the behaviour of a function converging at infinity. Similarly, a **regularly varying function** is a function of a real variable whose behaviour at infinity is similar to the behaviour of a power law function (like a polynomial) near infinity. These classes of functions were both introduced by Jovan Karamata,^{[1]}^{[2]} and have found several important applications, for example in probability theory.

Definition 1. A measurable function *L* : (0, +∞) → (0, +∞) is called *slowly varying* (at infinity) if for all *a* > 0,

Definition 2. Let *L* : (0, +∞) → (0, +∞). Then *L* is a regularly varying function if and only if . In particular, the limit must be finite.

These definitions are due to Jovan Karamata.^{[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).

Theorem 1. The limit in **definitions 1** and **2** is uniform if *a* is restricted to a compact interval.

Theorem 2. Every regularly varying function *f* : (0, +∞) → (0, +∞) is of the form

where

- β is a real number,
- L is a slowly varying function.

**Note**. 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*.

Theorem 3. A function *L* is slowly varying if and only if there exists *B* > 0 such that for all *x* ≥ *B* the function can be written in the form

where

*η*(*x*) is a bounded measurable function of a real variable converging to a finite number as*x*goes to infinity*ε*(*x*) is a bounded measurable function of a real variable converging to zero as*x*goes to infinity.

- If
*L*is a measurable function and has a limit

- then
*L*is a slowly varying function.

- For any
*β*∈**R**, the function*L*(*x*) = log^{ β}*x*is slowly varying. - The function
*L*(*x*) =*x*is not slowly varying, nor is*L*(*x*) =*x*^{ β}for any real*β*≠ 0. However, these functions are regularly varying.

- Analytic number theory
- Hardy–Littlewood tauberian theorem and its treatment by Karamata

- ^
^{a}^{b}^{c}See (Galambos & Seneta 1973) - ^
^{a}^{b}See (Bingham, Goldie & Teugels 1987).

- Bingham, N.H. (2001) [1994], "Karamata theory",
*Encyclopedia of Mathematics*, EMS Press - Bingham, N. H.; Goldie, C. M.; Teugels, J. L. (1987),
*Regular Variation*, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge: Cambridge University Press, ISBN 0-521-30787-2, MR 0898871, Zbl 0617.26001 - Galambos, J.; Seneta, E. (1973), "Regularly Varying Sequences",
*Proceedings of the American Mathematical Society*,**41**(1): 110–116, doi:10.2307/2038824, ISSN 0002-9939, JSTOR 2038824.