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On discrete limits of sequences of Darboux bilaterally quasicontinuous functions. (English) Zbl 1024.26002
According to Á.Császár and M. Laczkovich [Stud. Sci. Math. Hung. 10, 463-472 (1975; Zbl 0405.26006)], a function \(f:\mathbb{R}\to \mathbb{R}\) is the discrete limit of a sequence \(f_n: \mathbb{R}\to\mathbb{R}\) iff, for \(x\in\mathbb{R}\), there is \(n(x)\in\mathbb{N}\) such that \(f_n(x)= f(x)\) whenever \(n> n(x)\). \(f\) is bilaterally quasicontinuous at \(x\in\mathbb{R}\) iff, for \(\eta> 0\), there are nonempty open sets \(V\subset(x- \eta,x)\) and \(W\subset(x, x+\eta)\) such that \(f(V\cup W)\subset (f(x)- \eta, f(x)+ \eta)\) [see T. Neubrunn, Real Anal. Exch. 14, 259-306 (1989; Zbl 0679.26003)]. According to A. M. Bruckner [“Differentiation of real functions” (1978; Zbl 0382.26002; 2nd ed. 1994; Zbl 0796.26004)], \(x\in\mathbb{R}\) is a Darboux point of \(f\) iff, for \(r> 0\) and \(a\in (\min(f(x),\inf(K^+(f, x))), \max(f(x),\sup(K^+(f, x))))\) and \(b\in (\min(f(x), \inf(K^-(f, x))), \max(f(x),\sup(K^-(f, x))))\), there are points \(c\in (x, x+r)\) and \(d\in (x- r,x)\) such that \(f(c)= a\) and \(f(d)= b\), where \[ K^+(f, x)= \{y:\text{there is }x_n> x,\;x_n\to x\text{ with }f(x_n)\to y\} \] and \[ K^*(f, x)= \{y:\text{there is }x_n< x,\;x_n\to x\text{ with }f(x_n)\to y\}. \] The author proves that, if the set of all points where \(f\) is not bilaterally quasicontinuous and it is not Darboux is nowhere dense then \(f\) is the discrete limit of a sequence of Darboux bilaterally quasicontinuous functions. This result is sharp in a certain sense.
MSC:
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
26A21 Classification of real functions; Baire classification of sets and functions
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