Characterization of cliquish functions (Q5932743)

\textit{H. P. Thielman} defined cliquish functions in [Am. Math. Mon. 60, 156-161 (1953; Zbl 0051.13801)]. Let \((Y, d)\) be a metric space. A function \(f:X\to Y\) is said to be cliquish at \(x\in X\), if for each \(\varepsilon>0\) and each neighbourhood \(U\) of \(x\) there is an open non-empty set \(U_1\subset U\) such that \(d(f(x')\), \(f(x''))<\varepsilon\) for all \(x',x''\in U_1\). A function \(f\) is called cliquish if it is cliquish at each \(x\in X\). The main results of this paper are that for a real function \(f\) defined on a perfect Baire space \(X\) the following are equivalent: \(f\) is cliquish, and \(f\) is a pointwise limit of two monotone sequences of upper and lower quasi-continuous functions, respectively.