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1-improvable discontinuous functions. (English) Zbl 0885.26004
Assume that \(D\subset {\mathbb{R}}\) and \(f\colon D\to {\mathbb{R}}\) is a bounded function. A point \(x\in D\) is called an improvable point of discontinuity of \(f\) if: (1) there exists a limit \(\lim_{t\to x}f(t)\); (2) \(\lim_{t\to x}f(t)\neq f(x)\). (If \(x\) is an isolated point of \(D\) then \(\lim_{t\to x}f(t)=f(x)\).) Let \(U(f)\) denote the set of all improvable points of \(f\). Now “improve” \(f\) to the function \(f_{(1)}\) by putting \(f_{(1)}(x)=\lim_{t\to x}f(t)\) if \(x\in U(f)\) and \(f_{(1)}(x)=f(x)\) if \(x\not\in U(f)\). We say that \(f\) is 1-improvable if the function \(f_{(1)}\) is continuous [see A. Katafiasz, Real Anal. Exch. 21, No. 2, 407-423 (1995; Zbl 0879.26006); ibid. 21, No. 2, 430-439 (1995; Zbl 0879.26008)]. For \(D\) being a closed subset of \(\mathbb{R}\) the author characterizes the set \(C(f)\) of all continuity points of a 1-improvable function \(f: D\to {\mathbb{R}}\).

MSC:
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
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