00A20-Pathological.tex

\documentclass[12pt]{article}
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\pmcanonicalname{Pathological}
\pmcreated{2013-03-22 14:41:56}
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\pmtitle{pathological}
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\pmtype{Definition}
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\begin{document}
In mathematics, a \emph{pathological object} is mathematical
object that has a highly unexpected \PMlinkescapetext{property}. 

Pathological objects are typically percieved to, in some sense, be 
badly behaving. On the other hand, they are perfectly properly
defined mathematical objects. Therefore this ``bad behaviour'' can
simply be seen as a contradiction with our intuitive 
picture of how a certain object should behave. 

\subsubsection*{Examples}
\begin{itemize}
\item A very famous pathological function is the 
Weierstrass function, which is a continuous function 
that is nowhere differentiable. 
\item The Peano space filling curve. This pathological curve
maps the unit interval $[0,1]$ continuously onto $[0,1]\times [0,1]$. 
\item The Cantor set. This is subset of the interval $[0,1]$
has the pathological property that it is uncountable 
yet its measure is zero. 
\item The Dirichlet's function from $\R$ to $\R$ is continuous at every
irrational point and discontinuous at every rational point. 
\item Ackermann Function.
\end{itemize}

See also \cite{wiki}. 

\begin{thebibliography}{9}
\bibitem{wiki}Wikipedia \PMlinkexternal{entry on pathological, mathematics}{http://en.wikipedia.org/wiki/Pathological (mathematics)}. 
\end{thebibliography}
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