pathological
Pathological
2004-10-06 14:23:11
2009-01-12 12:07:38
Definition
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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}