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

Examples

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

See also [1].

Bibliography

1

Wikipedia entry on pathological, mathematics.