Pathological (mathematics)

Pathological examples are particular examples, which often occur in mathematical contexts. Definitions of mathematical objects are partially motivated by intuition, such as the definition of Wegzusammenhangs. In a pathological example, now an object is constructed which meets the conditions of a mathematically precise definition, however, is in conflict with the underlying view or for more evidence has undesirable properties, which are considered atypical for commonly occurring cases.

In the construction of pathological examples often the axiom of choice, recursive definitions and fractals are used.

Known pathological examples

Weierstrass functions

The Weierstrass function is continuous at every point, but nowhere differentiable. It is the first published example of such a function and changed the common opinion that every continuous function is differentiable except for a set of isolated points.

Dirichlet function

The Dirichlet function is at all rational points one and all irrational zero. It is an example of a function that is everywhere discontinuous and not Riemann integrable, but is Lebesgue integrable. A modification of the Dirichlet function is the thoma ash function. This function takes for irrational arguments to the value zero and a positive rational; In contrast to the Dirichlet function, this is Riemann integrable.

Cantor set

The Cantor set is a subset of the real numbers with special topological and measure theoretic properties. Thus, the amount is equally powerful as the set of real numbers, but it is also a Lebesgue -null set. Due to the same thickness, one might expect that quantities also have the same level. This is not the case, since the Lebesgue measure of the set of real numbers is infinite. As a topological space, the Cantor set is a compact, perfect, totally incoherent and nowhere dense subset of. Due to these properties, the Cantor set is used especially in the topology as an example, which often is contrary to the intuition.

Vitali amount

Vitali sets have the special property that one can assign them no Lebesgue measure. Not measurable quantities for the Lebesgue measure can be constructed using only the axiom of choice. But assuming this axiom can be constructed that solves the measurement problem not a measure. For other dimensions, however, it is easy to not show measurable quantities.