## 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.

### MSC:

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