The author presents a technique for expressing generating functions of sequences of integers defined using a real parameter \(x\), as for instance \(\sum_{n=1}^\infty z^{\lfloor n/x\rfloor}\), \(\sum_{n=1}^\infty \lfloor{n\over x}+1\rfloor\), etc.
The main theorem of the paper says: Let \(g(x,n)\), with \(x\in{\mathbb R}^+\), be a simple step complex valued function for each \(n\). Let \(\sum_{n=1}^\infty V_{(0,x)}(g(\cdot,n))<\infty\) for each \(x\), where \(V_I(g)\) denotes the variation of \(g\) on the interval \(I\). Let \(D\) be any set containing \(\{r; g(\cdot,n)\text{ is not continuous at }r\) for some \(n\)