Omega-limit sets and non-continuous functions (Q1978825)

This article is a summary of a talk given by the author concerning one-dimensional dynamics \((f,\mathbb{R})\). \(f\) represents a real-valued function on the real line here. The author studies omega-limit sets of quasicontinuous functions and, in particular, Darboux-Baire \(1\) (\(D\)-\(B_1\)) functions which satisfy Banach's condition \(T_2\). The author states the following results. Corollary 2: Let \(f:\mathbb{R}\rightarrow \mathbb{R}\) be a \(D\)-\(B_1\) function which satisfies Banach's condition \(T_2\). Then for any \(x\in \mathbb{R}\), the set \(\omega (x,f)\) is either nowhere dense, or is a finite union of nonsingleton connected closed sets. Theorem 2: Let \(f:\mathbb{R}\rightarrow \mathbb{R}\) be a \(D\)-\(B_1\) function, where \(I\) is a closed bounded interval. If \(J=[a,b]\subset I\) with \(f(J)\supset J\), then \(f\) has a fixed point in \(J\). Theorem 3: If \(f:\mathbb{R}\rightarrow \mathbb{R}\) is quasicontinuous, then for any \(x\in \mathbb{R}\), it follows that \(f({\text{Int}}(\omega (x,f))) \subset \omega (x,f)\).