Regulated functions. (English) Zbl 0724.26009

This paper considers the space of all regulated functions defined on a compact interval [a,b]. This space is, given the sup norm, a Banach space and a new characterization of the relatively compact sets is given. A set A of such functions is relatively compact iff the following conditions hold: (i) the set of functions has uniform unilateral limits at all relevant points of [a,b] (that is on [a,b[ for right limits, on ]a,b] for left limits); (ii) there is a \(\gamma =\gamma (t)\) such that for all relevant t and all \(x\in A\), \(| x(t)-x(t\pm)| \leq \gamma\); (iii) there is an \(\alpha\) such that for all \(x\in A\), \(| x(a)| \leq \alpha\). In addition an analogue of Helley’s (Principle of Choice) Theorem is proved.


26A45 Functions of bounded variation, generalizations
46E15 Banach spaces of continuous, differentiable or analytic functions
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