Pringsheim Convergence and the Dirichlet Function - Advances in Pure Mathematics

APM > Vol.6 No.6, May 2016

Pringsheim Convergence and the Dirichlet Function ()

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Download as PDF (Size: 277KB) PP. 441-445

DOI: 10.4236/apm.2016.66031 3,299 Downloads 5,762 Views

Author(s)

Thomas Beatty, Bradley Hansen

Affiliation(s)

Department of Mathematics, Florida Gulf Coast University, Ft. Myers, FL, USA.

ABSTRACT

Double sequences have some unexpected properties which derive from the possibility of commuting limit operations. For example,

may be defined so that the iterated limits

and

exist and are equal for all x, and yet the Pringsheim limit

does not exist. The sequence

is a classic example used to show that the iterated limit of a double sequence of continuous functions may exist, but result in an everywhere discontinuous limit. We explore whether the limit of this sequence in the Pringsheim sense equals the iterated result and derive an interesting property of cosines as a byproduct.

KEYWORDS

Convergence, Pointwise Limit, Double Sequence, Pringsheim, Dirichlet Function, Baire Category Theorem, Cosine

Share and Cite:

Beatty, T. and Hansen, B. (2016) Pringsheim Convergence and the Dirichlet Function. Advances in Pure Mathematics, 6, 441-445. doi: 10.4236/apm.2016.66031.

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