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Strong porosity features of typical continuous functions. (English) Zbl 0592.26010
A subset A of R is called bilaterally strongly porous at a point a, \(a\in R\), iff the right-porosity \(p_+(A,a)\) and the left-porosity \(p_- (A,a)\) of A at a are both 1. The right-porosity \(p_+(A,a)\) of A at a is the number: \(\limsup_{t\to 0+}\quad d(A,a,t)/t,\) where d(A,a,t) is the length of the largest open interval in \((x,x+t)-A.\) Their main result is the Theorem 2.5: For each \(\sigma\)-compact subset K of the space \(C(<0,1>)\) of all continuous functions on \(<0,1>\) with the supremum norm, the set \(F=\{f\in C(<0,1>):\) for each \(g\in K\) the set \(\{x\in <0,1>:\quad f(x)=g(x)\}\) is bilaterally strongly porous\(\}\) is residual in \(C(<0,1>)\). The authors give also three consequences of their result and concluding remarks. We give here their third consequence: For any system \(E=\{E_ x:\quad x\in <0,1>\}\) of paths, where \(E_ x\) is not bilaterally strongly porous at x for every \(x\in <0,1>\), the class of all functions of \(C(<0,1>)\) which are nowhere E-differentiable is an in \(C(<0,1>)\) residual set.
Reviewer: L.Mišík

MSC:
26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
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