Contingent

of a subset $E$ of a Euclidean space at a point $a\in E$

The union of rays $\vec{ab}$ with origin $a$ for which there exists a sequence of points $b_n\in E$ converging to $a$ such that the sequence of rays $\vec{ab_n}$ converges to $\vec{ab}$. It is denoted by $\operatorname{contg}(E,a)$. For an $m$-dimensional differentiable manifold $E$, $\operatorname{contg}(E,a)$ is the same as the $m$-dimensional tangent plane to $E$ at $a$. This concept proves useful in the study of differentiability properties of functions. If for every point $a$ of a set $E$ in the plane, $\operatorname{contg}(E,a)$ is not the whole plane, then $E$ 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 $p$-measure, $p=1,\dots,n-1$, located in an $n$-dimensional Euclidean space partitions into a countable number of parts, one of which has zero Favard measure of order $p$, while each of the remaining parts is situated on some Lipschitz surface of dimension $p$; for almost-all $x\in E$ (in the sense of the Hausdorff $p$-measure), $\operatorname{contg}(E,a)$ is a plane of dimension $p$ if all variations of the set $E$ are finite and, beginning with the $(p+1)$-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)