pathological
In mathematics, a pathological object is mathematical object that has a highly unexpected .
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.
Examples

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A very famous pathological function is the Weierstrass function^{}, which is a continuous function^{} that is nowhere differentiable^{}.

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The Peano space filling curve. This pathological curve maps the unit interval $[0,1]$ continuously onto $[0,1]\times [0,1]$.

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The Cantor set. This is subset of the interval $[0,1]$ has the pathological property that it is uncountable yet its measure is zero.

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The Dirichlet’s function from $\mathbb{R}$ to $\mathbb{R}$ is continuous at every irrational point and discontinuous^{} at every rational point.
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See also [1].
References
 1 Wikipedia http://en.wikipedia.org/wiki/Pathological (mathematics)entry on pathological, mathematics.
Title  pathological 

Canonical name  Pathological 
Date of creation  20130322 14:41:56 
Last modified on  20130322 14:41:56 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  10 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 00A20 