On monotonous separately continuous functions

Yaroslav I. Grushka

Ukraine

Institute of Mathematics NAS of Ukraine

Department of Nonlinear analysis

Submitted: 2018-03-12

|

Accepted: 2018-09-10

|

Published: 2019-04-01

DOI: https://doi.org/10.4995/agt.2019.9817
Funding Data

Downloads

Keywords:

separately continuous mappings, linearly ordered topological spaces, Young's theorem

Supporting agencies:

This research was not funded

Abstract:

Let T = (T, ≤) and T1= (T1 , ≤1) be linearly ordered sets and X be a topological space.  The main result of the paper is the following: If function ƒ(t,x) : T × X → T1 is continuous in each  variable (“t” and  “x”)  separately  and  function ƒx(t)  = ƒ(t,x) is  monotonous  on T for  every x ∈ X,  then ƒ is  continuous  mapping  from T × X to T1,  where T and T1 are  considered  as  topological  spaces  under  the order topology and T × X is considered as topological space under the Tychonoff topology on the Cartesian  product of topological spaces T and X.

Show more Show less

References:

G. Birkhoff, Lattice theory, Third edition. American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., New York, 1967.

K. C. Ciesielski and D. Miller, A continuous tale on continuous and separately continuous functions, Real Analysis Exchange 41, no. 1 (2016), 19-54. https://doi.org/10.14321/realanalexch.41.1.0019

O. Karlova, V. Mykhaylyuk and O. Sobchuk, Diagonals of separately continuous functions and their analogs, Topology Appl. 160, no. 1 (2013), 1-8. https://doi.org/10.1016/j.topol.2012.09.003

J. L. Kelley, General topology, University series in higher mathematics, Van Nostrand, 1955.

R. L. Krusee and J. J. Deely, Joint continuity of monotonic functions, The American Mathematical Monthly 76, no. 1 (1969), 74-76. https://doi.org/10.1080/00029890.1969.12000144

V. Mykhajlyuk, The Baire classification of separately continuous and monotone functions, Scientific Herald of Yuriy Fedkovych Chernivtsi National University 349 (2007), 95-97 (Ukrainian).

V. Nesterenko, Joint properties of functions which monotony with respect to the first variable, Mathematical Bulletin of Taras Shevchenko Scientific Society 6 (2009), 195-201 (Ukrainian).

H. Voloshyn, V. Maslyuchenko and O. Maslyuchenko, On layer-wise uniform approximation of separately continuous functioins by polynomials, Mathematical Bulletin of Taras Shevchenko Scientific Society 10 (2013), 135-158 (Ukrainian).

W. Young, A note on monotone functions, The Quarterly Journal of Pure and Applied Mathematics (Oxford Ser.) 41 (1910), 79-87.

Show more Show less