Some properties of a function connected to a double series (Q1872550)

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, \textit{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.