Displaying similar documents to “Nowhere monotone functions and a problem of K.M. Garg”

On Whitney pairs

Marianna Csörnyei (1999)

Fundamenta Mathematicae

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A simple arc ϕ is said to be a Whitney arc if there exists a non-constant function f such that    l i m x x 0 ( | f ( x ) - f ( x 0 ) | ) / ( | ϕ ( x ) - ϕ ( x 0 ) | ) = 0 for every x 0 . G. Petruska raised the question whether there exists a simple arc ϕ for which every subarc is a Whitney arc, but for which there is no parametrization satisfying    l i m t t 0 ( | t - t 0 | ) / ( | ϕ ( t ) - ϕ ( t 0 ) | ) = 0 . We answer this question partially, and study the structural properties of possible monotone, strictly monotone and VBG* functions f and associated Whitney arcs.

A Cantor set in the plane that is not σ-monotone

Aleš Nekvinda, Ondřej Zindulka (2011)

Fundamenta Mathematicae

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A metric space (X,d) is monotone if there is a linear order < on X and a constant c such that d(x,y) ≤ cd(x,z) for all x < y < z in X, and σ-monotone if it is a countable union of monotone subspaces. A planar set homeomorphic to the Cantor set that is not σ-monotone is constructed and investigated. It follows that there is a metric on a Cantor set that is not σ-monotone. This answers a question raised by the second author.

Connectedness in monotone spaces.

Ghosh, Saibal Ranjan, Dasgupta, Hiranmay (2004)

Bulletin of the Malaysian Mathematical Sciences Society. Second Series

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Monotone normality and extension of functions

Ian Stares (1995)

Commentationes Mathematicae Universitatis Carolinae

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We provide a characterisation of monotone normality with an analogue of the Tietze-Urysohn theorem for monotonically normal spaces as well as answer a question due to San-ou concerning the extension of Urysohn functions in monotonically normal spaces. We also extend a result of van Douwen, giving a characterisation of K 0 -spaces in terms of semi-continuous functions, as well as answer another question of San-ou concerning semi-continuous Urysohn functions.