The optimal form of selection principles for functions of a real variable

Let T be a nonempty set of real numbers, X a metric space with metric d and XT the set of all functions from T into X. If f∈XT and n is a positive integer, we set ν(n,f)=sup∑i=1nd(f(bi),f(ai)), where the supremum is taken over all numbers a1,…,an,b1,…,bn from T such that a1⩽b1⩽a2⩽b2⩽⋯⩽an⩽bn. The sequence {ν(n,f)}n=1∞ is called the modulus of variation of f in the sense of Chanturiya. We prove the following pointwise selection principle: If a sequence of functions{fj}j=1∞⊂XTis such that the closure in X of the set{fj(t)}j=1∞is compact for eacht∈Tand(∗)limn→∞(1nlim supj→∞ν(n,fj))=0,then there exists a subsequence of{fj}j=1∞, which converges in X pointwise on T to a functionf∈XTsatisfyinglimn→∞ν(n,f)/n=0. We show that condition (∗) is optimal (the best possible) and that all known pointwise selection theorems follow from this result (including Helly's theorem). Also, we establish several variants of the above theorem for the almost everywhere convergence and weak pointwise convergence when X is a reflexive separable Banach space.