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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.
Reviewer: M.Zelený (Praha)
##### 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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