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.[1] For a convex function that is not proper, there is disagreement as to the definition of the closure of the function.[citation needed]