Continuity points of typical bounded functions (Q1029994)

\textit{P. Kostyrko} and \textit{T. Šalát} [Real Anal. Exch. 10, 188--193 (1985; Zbl 0609.26002)] proved that if a linear space of bounded functions has an element that is discontinuous almost everywhere, then a typical element in the space is discontinuous almost everywhere. More precisely, they proved the following: If \(X\) is a linear subspace of the Banach space of all bounded functions such that \(\inf_{f\in X}\mu(C(f))\!=0\) (where \(\mu\) denotes the Lebesgue measure on [0,1]), then the set \(\{f\in X\mid \mu(C(f))=0\}\) is a residual \(G_{\delta}\) subset of \(X\). In the paper under review, the author proves the following topological analogue of this theorem: If \(X\) is a linear subspace of the Banach space of all bounded functions such that for each non-empty open subset \(U\) of [0,1] there exists an \(f\in X\) for which \(C(f)\) is not residual in \(U\), then the set \(\{f\in X\mid C(f)\) is nowhere dense\(\}\) is a residual \(G_{\delta}\) subset of \(X\). Some examples are discussed.