We investigate the class R ∞,s of rapidly varying sequences from different points of view. It is shown:(1) a sequence (c n ) n∈N of positive real numbers is rapidly varying if and only if the corresponding function x → c [x] , x 1, is rapidly varying; (2) the class R ∞,s satisfies some selection properties, as well as gametheoretical and Ramsey-theoretical conditions.

In this paper we consider the class of functions K c introduced by W. Matuszewska (1964, Studia Math. 24, 271-279) and W. Matuszewska and W. Orlicz (1965, Studia Math. 26, 11-24). As a main result we describe, in terms of the class K c , when two strictly increasing functions, as well as their inverse functions, are asymptoticaly equivalent in the strong sense. This result also gives a proper characterization of the class K c .

We present some results about selection properties in the class of double sequences of real numbers.

In this paper we prove that in the class of all measurable positive functions w . Ž . defined on the interval a, qϱ a ) 0 , the class of functions which preserve the Ä w . q Ž . strong asymptotic equivalence on the set of functions x: a, qϱ ¬ ‫ޒ‬ , x t ª 4 qϱ, t ª qϱ , is a class of O O-regularly varying functions with continuous index function. We also prove a representation theorem for functions from this class.

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