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Some properties of a function connected to a double series. (English) Zbl 1040.26003

Let \(X\) denote the set of all permutations of the set of positive integers endowed with the Fréchet metric \[ d(x,y)= \sum^\infty_{k=1} {1\over 2^k} {| x_k- y_k|\over 1+| x_k- y_k|}, \] where \(x= \{x_k\}^\infty_{k=1}\) and \(y= \{y_k\}^\infty_{k=1}\) are points of \(X\).
In 1956, H. M. Sengupta [Proc. Am. Math. Soc. 7, 347–350 (1956; Zbl 0074.04404)] studied some properties of a function defined on some subset of \(X\) relating \(x\) with a conditionally convergent series of real terms. This result led to define a function \(f\) on \(X\) into an interval related to a double series \(\sum_{m,n} a_{mn}\) of real terms as follows:
Let \(\sum a_{mn}\) be a non-absolutely convergent double series with \(a_{mn}\to 0\) as \(m,n\to\infty\) in Pringsheim’s sense. Then define \(f\) on \(X\) by \[ f(x)= {\sum_{m,n} \varepsilon_{mn}(x) a_{mn}\over 1+|\sum_{m,n} \varepsilon_{mn}(x) a_{mn}|} \] if \(\sum_{m,n} \varepsilon_{mn}(x) a_{mn}\) converges, otherwise zero, where \(\varepsilon_{mn}(x)\) takes the value \(0\) or \(1\) according to the integers corresponding to the position of terms are absent or present in the sequence of positive integers \(\{x_n\}^\infty_{n=1}\).
In this paper paper, the authors study a few properties of \(f: X\to (-1,1)\). First, they show that the set \[ \{x: x\in X,\;f(x)= \alpha\}, \] for each \(\alpha\in (-1,1)\), is dense in \(X\) and then the discontinuity of \(f\) everywhere. Further, they prove that \(f\) not only belongs to the third Borel class but it also has the Darboux property.

MSC:

26A21 Classification of real functions; Baire classification of sets and functions
40A05 Convergence and divergence of series and sequences

Citations:

Zbl 0074.04404
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