Une remarque sur les \(G_ \delta\)-denses
Let \(G\) be a dense \(G_ \delta\) subset of \(\mathbb{R}\) and \((f_ n)_{n\geq 0}\) a sequence of functions from \(\mathbb{R}\) to \(\mathbb{R}\), continuous and nowhere locally constant. By using Baire's theorem, the authors give an elementary proof that \(\bigcap_{n\geq 0} f^{-1}_ n(G)\) is also a \(G_ \delta\)-dense subset of \(\mathbb{R}\). By doing so, they generalize previous results of \textit{P. Erdős} [Michigan Math. J. 9, 59-60 (1962; Zbl 0114.26306)], \textit{G. J. Rieger} [Arch. Math. 26, 40-43 (1975; Zbl 0301.10029)] and \textit{W. Schwarz} [Math.-phys. Semesterber., n. F. 24, 84-87 (1977; Zbl 0344.10016)] on Liouville numbers.
scientific article; zbMATH DE number 590071