Closed convex function

In mathematics, a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is said to be closed if for each $\alpha \in \mathbb{R}$ , the sublevel set $\{ x \in \mbox{dom} f \vert f(x) \leq \alpha \}$ is a closed set.

Equivalently, if the epigraph defined by $\mbox{epi} f = \{ (x,t) \in \mathbb{R}^{n+1} \vert x \in \mbox{dom} f,\; f(x) \leq t\}$ 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.

单词	Closed convex function
释义	Closed convex function 英语百科 Closed convex function In mathematics, a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is said to be closed if for each $\alpha \in \mathbb{R}$ , the sublevel set $\{ x \in \mbox{dom} f \vert f(x) \leq \alpha \}$ is a closed set. Equivalently, if the epigraph defined by $\mbox{epi} f = \{ (x,t) \in \mathbb{R}^{n+1} \vert x \in \mbox{dom} f,\; f(x) \leq t\}$ 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.
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