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Journal of Function Spaces - Volume 2015 2015, Article ID 761924, 7 pages -

Research ArticleInstitute of Mathematics, Lodz University of Technology, Wólczańska 215, 93-005 Łódź, Poland

Received 30 October 2014; Accepted 14 January 2015

Academic Editor: Eva A. Gallardo Gutiérrez

Copyright © 2015 Artur Bartoszewicz and Szymon Głąb. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let be a family of continuous functions defined on a compact interval. We give a sufficient condition so that contains a dense -generated free algebra; in other words, is densely -strongly algebrable. As an application we obtain dense -strong algebrability of families of nowhere Hölder functions, Bruckner-Garg functions, functions with a dense set of local maxima and local minima, and nowhere monotonous functions differentiable at all but finitely many points. We also study the problem of the existence of large closed algebras within where or . We prove that the set of perfectly everywhere surjective functions together with the zero function contains a -generated algebra closed in the topology of uniform convergence while it does not contain a nontrivial algebra closed in the pointwise convergence topology. We prove that an infinitely generated algebra which is closed in the pointwise convergence topology needs to contain two valued functions and infinitely valued functions. We give an example of such an algebra; namely, it was shown that there is a subalgebra of with generators which is closed in the pointwise topology and, for any function in this algebra, there is an open set such that is a Bernstein set.

Author: Artur Bartoszewicz and Szymon Głąb



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