# Contingent

*of a subset of a Euclidean space at a point *

The union of rays with origin for which there exists a sequence of points converging to such that the sequence of rays converges to . It is denoted by . For an -dimensional differentiable manifold , is the same as the -dimensional tangent plane to at . This concept proves useful in the study of differentiability properties of functions. If for every point of a set in the plane, is not the whole plane, then can be partitioned into a countable number of parts situated on rectifiable curves. This theorem has been repeatedly generalized and refined. For example, a set of finite Hausdorff -measure, , located in an -dimensional Euclidean space partitions into a countable number of parts, one of which has zero Favard measure of order , while each of the remaining parts is situated on some Lipschitz surface of dimension ; for almost-all (in the sense of the Hausdorff -measure), is a plane of dimension if all variations of the set are finite and, beginning with the -th, vanish.

#### References

[1] | G. Bouligand, "Introduction à la géometrie infinitésimale directe" , Vuibert (1932) |

[2] | S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French) |

[3] | H. Federer, "Geometric measure theory" , Springer (1969) |

[4] | L.D. Ivanov, "Variations of sets and functions" , Moscow (1975) (In Russian) |

#### Comments

More on contingents (and the related notion of paratingent) can be found in G. Choquet's monograph [a1]. Contingents are useful in optimization problems nowadays.

#### References

[a1] | G. Choquet, "Outils topologiques et métriques de l'analyse mathématiques" , Centre Docum. Univ. Paris (1969) (Rédigé par C. Mayer) |

[a2] | J.P. Aubin, I. Ekeland, "Applied nonlinear analysis" , Wiley (Interscience) (1984) |

**How to Cite This Entry:**

Contingent.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Contingent&oldid=12121