## 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 FrĂ©chet 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

### Keywords:

rearrangement; Baire class; Borel class; Darboux property

Zbl 0074.04404
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