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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
##### Keywords:
improvable point of discontinuity; improvable function
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