Difference functions for functions with the Baire property
Let \(G\) be the additive group \(\mathbb R\) or \(\mathbb R/\mathbb Z\). For \(f:G\to\mathbb R\) and \(h\in G\) the difference function is defined by \(\Delta_hf(x)=f(x+h)-f(x)\). For families \(\mathcal G\subset\mathcal F\) of functions \(f:G\to\mathbb R\) let \(\mathcal H(\mathcal F,\mathcal G)= \{H\subset G:\exists f\in\mathcal F\smallsetminus\mathcal G\) \(\forall h\in H\) \(\Delta_hf\in\mathcal G\}\). \textit{T. Keleti} in his thesis studied such families for several families \(\mathcal F\), \(\mathcal G\) of measurable functions. In the paper under the review the authors prove analogous results for families of functions with the Baire property. In particular, for \(\mathcal G\) the family of functions with the Baire property is considered and for \(\mathcal F\) two of its subfamilies, the family of functions which are almost everywhere equal to a continuous function and the family of functions which are almost everywhere equal to a bounded function (``almost everywhere'' here means with respect to the first category).
scientific article; zbMATH DE number 1320502