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单词 Pathological
释义

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 contradictionMathworldPlanetmathPlanetmath with our intuitivepicture of how a certain object should behave.

Examples

  • A very famous pathological function is theWeierstrass functionMathworldPlanetmath, which is a continuous functionMathworldPlanetmathPlanetmaththat is nowhere differentiableMathworldPlanetmathPlanetmath.

  • The Peano space filling curve. This pathological curvemaps the unit interval [0,1] continuously onto [0,1]×[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 to is continuous at everyirrational point and discontinuousMathworldPlanetmath at every rational point.

  • Ackermann FunctionMathworldPlanetmath.

See also [1].

References

  • 1 Wikipedia http://en.wikipedia.org/wiki/Pathological (mathematics)entry on pathological, mathematics.
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更新时间:2025/5/4 12:22:28