Closed convex function

In mathematics, a function f : R n → R is said to be closed if for each α ∈ R , the sublevel set { x ∈ dom f | f ( x ) ≤ α } is a closed set.

Equivalently, if the epigraph defined by epi f = { ( x , t ) ∈ R n + 1 | x ∈ dom f , f ( x ) ≤ t } is closed, then the function f ( x ) 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.