Fine tuning the recoverability of Baire one functions. (English) Zbl 0854.26002

Evans, Michael J.; O’Malley, Richard J.

Real Anal. Exch. 21(1995-96), No. 1, 165-174 (1996).

Any sequence \(\{x_n\}^\infty_{n = 0}\) of distinct points in \([0,1]\) which is dense in \([0,1]\) is said to be a trajectory. Let \(\{x_n\}\) be a fixed trajectory. For a given interval, or finite union of intervals, \(H \subset [0,1]\), \(r(H)\) will be the first element of the trajectory \(\{x_n\}\) in \(H\). Let \(x \in [0,1]\), \(\rho > 0\) and \(B_\rho(x) = \{y \in [0,1] : |x - y|< \rho\}\). The first return route to \(x\), \(R_x = \{y_k\}^\infty_{k = 1}\), is defined recursively via \(y_1 = x_0\), \(y_{k+1} = r(B_{|x- y_k|} (x))\) if \(x \neq y_k\) and \(y_{k+1} = y_k\) if \(x = y_k\). A function \(f : [0,1] \to R\) \((R\) – the real line) is first return recoverable with respect to \(\{x_n\}\) at \(x\) provided that \(\lim_{k \to \infty} f(y_k) = f(x)\), and if this happens for each \(x \in [0,1]\), then \(f\) is first return recoverable with respect to \(\{x_n\}\). For \(0 < x \leq 1\) the left first return path to \(x\) based on \(\{x_n\}\), \(P^1_x = \{t_k\}\), is defined recursively via \(t_1 = r(0,x)\), and \(t_{k + 1} = r(t_k, x)\) and for \(0 \leq x < 1\) the right first return path to \(x\) based on \(\{x_n\}\), \(P^r_x = \{s_k\}\), is defined analogously. The function \(f : [0,1] \to R\) is first return continuous at \(x \in (0,1)\) with respect to the trajectory \(\{x_n\}\) provided \(\lim_{t \to x, t\in P^1_x} f(t) = f(x)\) and \(\lim_{s\to x, s\in P^r_x} f(s) = f(x)\). Analogously, for each \(x \in [0,1]\) the first return approach to \(x\) based on \(\{x_n\}\), \(A_x = \{u_k\}\), is defined recursively via \(u_1 = r((0,1) - \{x\})\), and \(u_{k+1} = r(B_{|x - u_k|} (x) - \{x\})\). \(f\) is first return approachable at \(x\) with respect to the trajectory \(\{x_n\}\) provided \(\lim_{u \to x, u\in A_x} f(u) = f(x)\).
The authors categorize the points \(x\) into types depending upon how isolated \((x,f(x))\) is in the graph of \(f\): Type I points are those \(x \in (0,1)\) for which \((x,f(x))\) is isolated on neither the left nor the right. Type II points are those \(x \in (0,1)\) for which \((x,f(x))\) is isolated on exactly one side. The main result of the paper follows: Let \(f : [0,1] \to R\) be a Baire one function. Then there exists a trajectory \(\{x_n\}\) such that a) \(f\) is first return recoverable with respect to \(\{x_n\}\); b) If \(x\) is a type I point, then \(f\) is first return continuous at \(x\) with respect to \(\{x_n\}\); c) If \(x\) is a type II point, then \(f\) is first return approachable at \(x\) with respect to \(\{x_n\}\).

Reviewer: P.Kostyrko (Bratislava)

MSC:

26A03	Foundations: limits and generalizations, elementary topology of the line
26A21	Classification of real functions; Baire classification of sets and functions
26A15	Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable

Keywords:

trajectory; first return recoverable; first return continuous; Baire one function