Analogues of the Denjoy-Young-Saks theorem

Belna, C. L.; Cargo, G. T.; Evans, Michael J.; Humke, Paul D.

doi:10.1090/s0002-9947-1982-0648091-0

Cited by 9 publications

(9 citation statements)

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“…This implies that the set of points at which both u, and l¡ are continuous (as extended real-valued functions) is residual. As was shown in the beginning of the proof of Theorem 3.2 the inequalities (1) uf(x)>f(x)>lf(x) are true on a residual set. This implies that the set of all points at which (1) is true and both uf and lf are continuous is residual.…”

Section: X^x0mentioning

confidence: 68%

“…Let A C R and x E R. We define the reflection of A about x to be the set RX(A)= {2x-t:t EA}. E. P. Dolzenko [6] has introduced the notion of porosity and it has recently found application in the study of differentiation; e.g., see [1,2,21]. The porosity of the set A at the point x E R is defined to be limr_0+ l(x, r, A)/r where l(x, r, A) denotes the length of the largest open interval contained in the set (x -r, x + r) H Ac.…”

Section: X^x0mentioning

confidence: 99%

“…In the version of Theorem 3.4 for ordinary dérivâtes (Theorem 2 in [1] or Theorem 1 in [21]) the exceptional set D is a-porous. That this need not be the case in the qualitative setting, even for functions in ß, is seen by considering the characteristic function / of a first category set S whose complement has measure zero.…”

Section: X^x0mentioning

confidence: 99%

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Qualitative differentiation

Evans¹,

Larson²

1983

Trans. Amer. Math. Soc.

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Abstract. Qualitative dérivâtes and derivatives, as well as qualitative symmetric dérivâtes and derivatives, are studied in the paper. Analogues of several results known for ordinary dérivâtes and derivatives are obtained in the qualitative setting.1. Introduction. The notions of qualitative limits, qualitative continuity, and qualitative derivatives were introduced by S. Marcus [13][14][15]. The purpose of the present paper is to examine qualitative differentiation and qualitative symmetric differentiation and, in particular, to present analogues of results known to hold for ordinary differentiation, symmetric differentiation, approximate differentiation, and approximate symmetric differentiation. Loosely speaking, qualitative differentiation and qualitative symmetric differentiation may be thought of as category analogues of approximate differentiation and approximate symmetric differentiation, where the set neglected near a point in the computation of difference quotients is of first category at the point in the former setting instead of density zero at the point as in the latter.We state our definitions in §2. In §3 we examine qualitative derivatives and dérivâtes. There we show that a qualitatively differentiable function on the real line is actually differentiable everywhere and obtain what may be viewed as qualitative analogues of the Denjoy-Young-Saks theorem [18]. In §4 we consider qualitative symmetric derivatives and dérivâtes. We show that with mild continuity restrictions on the primitive, a qualitative symmetric derivative must belong to Baire class one and actually be the symmetric derivative of a closely related function except at countably many points. A monotonicity theorem and related results are given.

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Section: X^x0mentioning

confidence: 68%

Section: X^x0mentioning

confidence: 99%

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Qualitative differentiation

Evans¹,

Larson²

1983

Trans. Amer. Math. Soc.

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“…differentiable. In fact, Belna, Cargo, Evans and Humke [3] show that there exists a strictly increasing homeomorphism h : We are not able to find a satisfactory answer to our question. Therefore, the following remains.…”

Section: On the Other Handmentioning

confidence: 76%

“…Now suppose that J contains no interval and that f, K are as in (ii) such that f (z n ) → G(z). Hence, by (3), G(z) is finite, so g(z) = G(z) and |g(z) − g(x)| ≤ ε/2 < ε. Therefore, g is continuous at x.…”

Section: A Simple Category Topologymentioning

confidence: 99%

ℐ-density continuous functions

Ciesielski¹,

Larson²,

Ostaszewski³

1994

Memoirs of the AMS

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The I-density topology is a generalization of the ordinary density topology to the setting of category instead of measure. This work involves functions which are continuous when combinations of the I-density, deep-I-density, density and ordinary topology are used on the domain and range. In the process of examining these functions, the I-density and deep-I-density topologies are deeply explored and the properties of these function classes as semigroups are considered.

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References

2013

The Elements of Cantor Sets

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Analogues of the Denjoy-Young-Saks theorem

Cited by 9 publications

References 10 publications

Qualitative differentiation

Qualitative differentiation

ℐ-density continuous functions

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