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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