%0 Journal Article
%T Pringsheim Convergence and the  Dirichlet Function
%A Thomas Beatty
%A Bradley Hansen
%J Advances in Pure Mathematics
%P 441-445
%@ 2160-0384
%D 2016
%I Scientific Research Publishing
%R 10.4236/apm.2016.66031
%X Double sequences have some unexpected properties which derive from the
possibility of commuting limit operations. For example, <img src=\"Edit_f8e89dfa-b891-40b7-bde7-c051aab8e013.bmp\" alt=\"\" /><span style=\"font-size:10.0pt;font-family:;\" \"=\"\">may be defined so that the iterated limits <img src=\"Edit_2c50b1f3-0479-4558-a31b-94dd28c87aae.bmp\" alt=\"\" /> </span><span style=\"font-size:10.0pt;font-family:;\" \"=\"\">and <img src=\"Edit_9187b87c-401d-4de7-b741-77a7e6d4fcb0.bmp\" alt=\"\" />&nbsp;</span><span style=\"font-size:10.0pt;font-family:;\" \"=\"\">exist and are equal for all <i>x</i>, and yet the Pringsheim limit <img src=\"Edit_8251837f-dbea-4152-b7bf-1cbb5c7416e3.bmp\" alt=\"\" /> </span><span style=\"font-size:10.0pt;font-family:;\" \"=\"\">does not exist. The
sequence <img src=\"Edit_baf0e649-a818-49a9-8891-3840321b70af.bmp\" alt=\"\" /></span><span style=\"font-size:10.0pt;font-family:;\" \"=\"\">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.</span>
%K Convergence
%K Pointwise Limit
%K Double Sequence
%K Pringsheim
%K Dirichlet Function
%K Baire Category Theorem
%K Cosine
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=66688