×

Porosity and typical properties of real-valued continuous functions. (English) Zbl 0708.26002

Let C be the space of all real-valued continuous functions on [0,1]. We say that nearly all elements of C have a certain property if those not enjoying it form a \(\sigma\)-porous set. The function \(f\in C\) is said to be of monotonic type at x if there is a real number c such that the function \(t\to f(t)-ct\) is monotonic at x. It is proven: 1) Nearly all elements of C are of nonmonotonic type at any x. 2) For nearly all elements f of C at each point \(x\in [0,1):\) \(D^+f(x)=\infty\) or \(D_+f(x)=-\infty\), and at each point \(x\in (0,1]:\) \(D^-f(x)=\infty\) or \(D_ -f(x)=-\infty\). 3) For nearly all elements f of C at most points \(x\in [0,1]:\) \(D_ -f(x)=D_+f(x)=-\infty\) and \(D^- f(x)=D^+f(x)=\infty\).
Reviewer: Z.Grande

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
26A12 Rate of growth of functions, orders of infinity, slowly varying functions
26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
54C30 Real-valued functions in general topology
54C50 Topology of special sets defined by functions
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Banach, S., Über die Bairesche Kategorie gewisser Funktionenmengen, Studia Math., 3, 174-179 (1931) · Zbl 0003.29703
[2] A. M. Bruckner, Differentation of Real Functions, L.N.M. 659, Berlin-Heidelberg-New York 1978. · Zbl 0382.26002
[3] Dolženko, E. P., The boundary properties of arbitrary functions, Russian, Izv. Akad. Nauk. SSSR, Ser. Mat., 31, 3-14 (1967) · Zbl 0177.10604
[4] Jarník, V., Über die Differenzierbarkeit stetiger Funktionen, Fund. Math., 21, 48-58 (1933) · JFM 59.0287.03
[5] Mazurkiewicz, S., Sur les fonctions non-dérivables, Studia Math., 3, 92-94 (1931) · Zbl 0003.29702
[6] Neugebauer, C., A Theorem on derivatives, Acta Sci. Math. (Szeged), 23, 79-81 (1962) · Zbl 0105.04602
[7] L. Zají Ček, Sets of cr-porosity and sets of σ-porosity (q). Casopis Pest. Mat. 101 (1976). · Zbl 0341.30026
[8] Zamfirescu, T., How many sets are porous?, Proc. Amer. Math. Soc., 100, 383-387 (1987) · Zbl 0625.54036 · doi:10.2307/2045976
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.