TITLE:
Pringsheim Convergence and the Dirichlet Function
AUTHORS:
Thomas Beatty, Bradley Hansen
KEYWORDS:
Convergence, Pointwise Limit, Double Sequence, Pringsheim, Dirichlet Function, Baire Category Theorem, Cosine
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.6 No.6,
May
24,
2016
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.