zbMATH — the first resource for mathematics

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

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
Full Text: DOI
[1] S. Banach, Über die Baire’sche Kategorie gewisser Funcktionenmengen,Studia Math.,3 (1931), 174–179. · JFM 57.0305.05
[2] A. Bruckner, Current trends in differentiation theory,Real Analysis Exchange,5 (1979–80), 9–60. · Zbl 0446.26004
[3] A. Bruckner and K. Garg, The level structure of a residual set of continuous functions,Trans. Amer. Math. Soc.,232 (1977), 307–321. · Zbl 0372.46027
[4] A. Bruckner, J. Ceder, and M. Weiss, On the differentiability structure of real functions,Trans. Amer. Math. Soc.,142 (1969). · Zbl 0182.38301
[5] A. Bruckner, R. O’Malley, and B. Thomson, Path derivatives: a unified view of certain generalized derivatives,Trans. Amer. Math. Soc.,283 (1984), 97–125. · Zbl 0541.26003
[6] C. Goffman and G. Pedrick,First Course in Functional Analysis, Prentice-Hall, Inc., 1965. · Zbl 0122.11206
[7] V. Jarnik, Sur dérivabilité des fonctions continues,Spisy Privodov, Fak. Univ. Karlovy,129 (1934), 3–9.
[8] V. Jarnik, Über die Differenzierbarkeit stetiger Funktionen,Fund. Math.,21 (1933), 48–58. · Zbl 0007.40102
[9] V. Jarnik, Sur les nombers dérivées approximatifs,Fund. Math.,22 (1934), 4–16.
[10] M. Laczkovich, Differentiable restrictions of continuous function, to appear. · Zbl 0558.26005
[11] J. Marcinkiewicz, Sur les nombres dérivée,Fund. Math.,24 (1935), 304–308. · Zbl 0011.10705
[12] S. Mazurkiewicz, Sur les fonctions non-dérivable,Studia Math.,3 (1931), 92–94. · Zbl 0003.29702
[13] J. Scholz, Essential derivations of functions inC, (to appear).
[14] B. Thomson, On the level set structure of a continuous function, to appear. · Zbl 0568.26001
[15] L. Zajícek, On Dini derivatives of continuous and monotone functions,Real Anal7sis Exchange,7 (1981–82), 233–238.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.