Closed convex function
In mathematics, a function is said to be closed if for each, the sublevel set
is a closed set.
Equivalently, if the epigraph defined by
is closed, then the function is closed.
This definition is valid for any function, but most used for convex functions. A proper [convex function] is closed if and only if it is lower semi-continuous.
Properties
- If is a continuous function and is closed, then is closed.
- If is a continuous function and is open, then is closed if and only if it converges to along every sequence converging to a boundary point of.
- A closed proper convex function f is the pointwise supremum of the collection of all affine functions h such that h ≤ f.