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“More or less” first-return recoverable functions. (English) Zbl 1088.26002

Let \(B_{\rho }(x)=\{y\in [0,1]\: | y-x| <\rho \}\) and let \((x_n)\) be a sequence of distinct points of [0,1] dense in [0,1]. Denote by \(r\bigl (B_{\rho }(x)\bigr )\) the first element of the trajectory \((x_n)\) in \(B_{\rho }(x)\). The first route to \(x\), \(R_x=(w_k)\), is defined recursively via \(w_1=x_1\), \(w_{k+1}=r\bigl (B_{| x-w_k| }(x)\bigr )\) if \(x\neq w_k\) or \(x\) if \(x=w_k\). A function \(f\: [0,1]\to \mathbb R\) is said to be the first return recoverable with respect to \((x_n)\) at \(x\) if \(\lim _{k\to \infty } f(w_k)=f(x)\). In [U. B. Darji, M. J. Evans and R. J. O’Malley, Real Anal. Exch. 19, No. 2, 510–515 (1994; Zbl 0840.26005)] it is shown that \(f\) is first return recoverable for some trajectory \((x_n)\) at each point if and only if \(f\) is Baire one.
In this paper, the authors pursue a systematic investigation of classifications of functions which arise when strengthening (universal recoverability and consistent recoverability) and weakening (except for points of a measure zero, for a set of the first category, for a countable set, and for a scattered set) the notion of the first recovery.
Let us mention one typical result: A function \(f\) belongs to the Baire one class and the set of all non-quasicontinuity points is countable if and only if there is a countable set \(S\) such that every countable set \(D\) in [0,1] has an ordering which recovers \(f\) at each point of \([0,1]\setminus S\).

MSC:

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

Citations:

Zbl 0840.26005
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