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Porosity, derived numbers and knot points of typical continuous functions. (English) Zbl 0672.26002
In this paper, the author improved the result of V. Jarník (1934) that there exists a residual subset A of $$C(<0,1>)$$ such that the set of all knot points in $$<0,1>$$ of f is of full measure, whenever $$f\in A.$$ The author’s Theorem 1 sets one of the B. S. Thomson’s theorem on Dini derivates of continuous functions in a more general context. The main theorem of the paper is the Theorem 2.
Let G be the set of all increasing functions $$g: (0,\infty)\to (0,\infty)$$ such that $$g(x)>x$$ holds for all $$x\in (0,\infty).$$ Let be $$g\in G,$$ $$A\subset R$$ and $$a\in R:$$ the set A is [g]-porous from the right at a iff there exists a decreasing sequence $$\{h_ n\}^{\infty}_{n=1}$$ of positive numbers tending to zero such that $$g(p(A,(a,a+h_ n)))>h_ n$$ for all n; $$p(A,(a,a+h_ n))$$ is the maximal length of an interval $$I\subset (a,a+h_ n)$$ missing the set A.
Let f be a real function defined on $$<0,1>$$, $$g\in G$$, $$a\in <0,1>$$ and $$c\in R\cup \{-\infty,\infty \}.$$ Then c is called a right [g]-derived number of f at a iff the set $$\{x\in <0,1>-\{a\}:\quad (f(x)-f(a))/(x- a)\not\in V\}$$ is [g]-porous from the right at a whenever V is a neighbourhood in $$R\cup \{-\infty,\infty \}$$ of c. A point a is called a [g]-knot point of f iff every $$c\in R\cup \{-\infty,\infty \}$$ is a bilateral [g]-derived number of f at a. The set $$D=\{<a+nh,a+(n+1)h>:$$ n is an integer$$\}$$ is called equidistant division of R with the norm h. A subset A of R is called [g]-totally porous iff for every $$\epsilon >0$$ there exists an equidistant division D of R with the norm less than $$\epsilon$$, such that $$g(p(A,I))>$$ than the length of I for any $$I\in D.$$
Theorem 2. For any $$g\in G$$ there exists a residual subset E of $$C(<0,1>)$$ such that the set $$\{a\in <0,1>:$$ a is not a [g]-knot point of $$f\}$$ is a $$\sigma$$-[g]-totally porous set for every $$f\in E.$$
At the end of the paper there are formulated two problems. The author added in a footnote, that the answer to the second problem is negative.
Reviewer: L.Mišík
##### MSC:
 26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems 26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
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##### References:
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