1 Answers
In mathematics, a function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} } is said to be closed if for each α ∈ R {\displaystyle \alpha \in \mathbb {R} } , the sublevel set { x ∈ dom f | f ≤ α } {\displaystyle \{x\in {\mbox{dom}}f\vert f\leq \alpha \}} is a closed set.
Equivalently, if the epigraph defined by epi f = { ∈ R n + 1 | x ∈ dom f , f ≤ t } {\displaystyle {\mbox{epi}}f=\{\in \mathbb {R} ^{n+1}\vert x\in {\mbox{dom}}f,\;f\leq t\}} is closed, then the function f {\displaystyle f} 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.