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Averages of quasi-continuous functions. (English) Zbl 0936.26001
The author proves the following result: Let \(\mathcal F\) be one of the following classes of functions from \(\mathbb R\) to \(\mathbb R\): all cliquish functions, Lebesgue measurable cliquish functions, cliquish functions in Baire class \(\alpha \) (\(\alpha\geq 1\)), and suppose \(f_1,\dots,f_k\in \mathcal F\). Then the following properties are equivalent:
(a) there is a positive function \(g\) such that \(f_1 + g,\dots,f_k+g\) are quasi-continuous,
(b) there is a positive function \(g \in \mathcal F\) such that \(\mathcal C(g) \supset \bigcap _{i=1}^k\mathcal C(f_i)\) and \(f_1 + g, \dots , f_k+g\) are
{(b)} quasi-continuous (\(\mathcal C(f)\) denotes the set of points of continuity of \(f\)),
(c) for each \(x \in \mathbb R\) and each \(i=1,\dots,k\) we have \[ \liminf _{t\to x,\;t\in\mathcal C(f_i)}f_i(t)<\infty. \] A similar result concerning Darboux quasi-continuous functions instead of quasi-continuous ones is also presented.
MSC:
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
54C08 Weak and generalized continuity
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