pathological
In mathematics, a pathological object is mathematicalobject that has a highly unexpected .
Pathological objects are typically percieved to, in some sense, bebadly behaving. On the other hand, they are perfectly properlydefined mathematical objects. Therefore this “bad behaviour” cansimply be seen as a contradiction with our intuitivepicture of how a certain object should behave.
Examples
- •
A very famous pathological function is theWeierstrass function
, which is a continuous function
that is nowhere differentiable
.
- •
The Peano space filling curve. This pathological curvemaps the unit interval continuously onto .
- •
The Cantor set. This is subset of the interval has the pathological property that it is uncountableyet its measure is zero.
- •
The Dirichlet’s function from to is continuous at everyirrational point and discontinuous
at every rational point.
- •
Ackermann Function
.
See also [1].
References
- 1 Wikipedia http://en.wikipedia.org/wiki/Pathological (mathematics)entry on pathological, mathematics.