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On a typical property of functions. (English) Zbl 0835.26005
Let $$s$$ be the space of all real sequences endowed with the Fréchet metric $$\rho$$, $\rho(a, b)= \sum^\infty_{k= 1} 2^{- k}|a_k- b_k|/(1+ |a_k- b_k|),\;a= \{a_k\}_k,\;b= \{b_k\}_k\in s.$ Let $$\mathcal F$$ denote the space of all functions $$f: \mathbb{R}\to \mathbb{R}$$ with the metric of uniform convergence. A set $$E\subset s$$ is said to be superporous at $$a\in s$$, if $$E\cup F$$ is porous at $$a$$ whenever $$F\subset s$$ is porous at $$a$$. The set $$E$$ is said to be superporous if it is superporous at each of its points and $$E$$ is said to be $$\sigma$$-superporous if it is a countable union of superporous sets.
The authors deal with the class $$\mathcal U$$ of all functions $$f\in {\mathcal F}$$ for which the set {$$\{a_k\}_k\in s: \sum_k f(a_k)$$ converges} is $$\sigma$$-superporous in $$(s, \rho)$$. They show that $$\mathcal U$$ is residual in $$\mathcal F$$, both $$\mathcal U$$ and $${\mathcal F}- {\mathcal U}$$ are dense-in-itself and $$\mathcal U$$ is a Baire space in the relative topology.

##### MSC:
 26A21 Classification of real functions; Baire classification of sets and functions 40A05 Convergence and divergence of series and sequences 54E52 Baire category, Baire spaces
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##### References:
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