Porosity, derived numbers and knot points of typical continuous functions.

*(English)*Zbl 0672.26002In 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.

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 |

##### Keywords:

typical continuous functions; knot points; Dini derivates of continuous functions; derived number; porous set
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\textit{L. Zajíček}, Czech. Math. J. 39(114), No. 1, 45--52 (1989; Zbl 0672.26002)

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##### References:

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