Variational analysis

thumb|right|An upper semicontinuous function that is not lower semicontinuous at x_0. The solid blue dot indicates f\left(x_0\right). thumb|right|A lower semicontinuous function that is not upper semicontinuous at x_0. The solid blue dot indicates f\left(x_0\right).

commutative associative binary operation on subsets of an Abelian group (such as Euclidean space)

the set of points lying on or above the graph of a function

right|thumb|A convex function (blue) and "subtangent lines" at x_0 (red). In mathematics, the subderivative (or subgradient) generalizes the derivative to convex functions which are not necessarily differentiable. The set of subderivatives at a point is called the subdifferential at that point. Subderivatives arise in convex analysis, the study of convex functions, often in connection to convex optimization.

concept in calculus of variation