On the structure of some function space (Q1086688)

Let \(\lambda\) denote Lebesgue measure and \(D(f)\) be the set of discontinuities of \(f\). The following theorem is proved: Let \(F\) be a linear space of bounded functions \(f:[0,1]\to \mathbb R\) furnished with the sup norm. If for each \(t<1\) there exists an \(h\in F\) such that \(\lambda (D(h))>t,\) then the subclass \(\{f\in F: \lambda (D(f))=1\}\) is a residual \(G_{\delta}\) subset. The result is applied to a number of specific classes of functions.