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. For a convex function which is not proper there is disagreement as to the definition of the closure of the function. 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 (called the affine minorants of f).