Locally bounded functions
Let \(X\) be a topological space, \(M\) a metric space. A function \(f:X\to M\) is called locally bounded if for any \(x\in X\) there exists an open set~\(U\) containing~\(x\) such that \(f\) is bounded on~\(U\). It is easy to see that any continuous function is locally bounded. Two properties of functions whose codomains are topological spaces are discussed: Fuller's subcontinuity and local compactness. For real-valued functions on first countable Hausdorff spaces, subcontinuity is equivalent to local boundedness. If \(X\) is locally compact and \(Y\) is completely regular, then a function \(f:X\to Y\) is subcontinuous iff \(f\) is locally compact. Functions with at worst removable discontinuities are locally bounded. If \(X\) is a compact space and \(f:X\to \mathbf R\) is locally bounded, then \(f\) is bounded. The space \({\mathcal LB}(X,M)\) of all locally bounded functions from~\(X\) into~\(M\) endowed with the compact-open topology is \(T_{2{1\over 2}}\) but need not be regular. If \(X\) is homeomorphic to~\(Y\), then \({\mathcal LB}(X,M)\) is homeomorphic to \({\mathcal LB}(Y,M)\).
scientific article; zbMATH DE number 1449417