We are going to prove that level sets of continuous functions increasing with respect to each variable are arcwise connected (Theorem 3) and characterize those of them which are arcs (Theorem 2). In [3], we will apply the second result to the classical linear functional equation φ∘f = gφ + h (cf., for instance, [1] and [2]) in a case not studied yet, where f is given as a pair of means, that is so-called mean-type mapping.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1056,
author = {Katarzyna Sajbura},
title = {Level sets of continuous functions increasing with respect to each variable},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
volume = {25},
year = {2005},
pages = {19-26},
zbl = {1170.26302},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1056}
}
Katarzyna Sajbura. Level sets of continuous functions increasing with respect to each variable. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 25 (2005) pp. 19-26. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1056/
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[001] [2] M. Kuczma, B. Choczewski and R. Ger, Iterative functional equations, Encyclopedia of mathematics and its applications 32, Cambridge University Press, Cambridge, 1990. | Zbl 0703.39005
[002] [3] K. Sajbura, On a linear functional equation with a mean-type mapping having no fixed points, Discuss. Math. DICO 25 (2005), 27-46. | Zbl 1111.39019