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 functionthat is nowhere differentiable.

• The Peano space filling curve. This pathological curvemaps 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 uncountableyet its measure is zero.

• The Dirichlet’s function from $ℝ$ to $ℝ$ is continuous at everyirrational point and discontinuous at every rational point.

• Ackermann Function.

See also [1].