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Closed convex function - Wikipedia

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In mathematics, a function 𝑓 : 𝑅 𝑛 β†’ 𝑅 is said to be closed if for each 𝛼 ∈ 𝑅 , the sublevel set { π‘₯ ∈ dom 𝑓 | 𝑓 ( π‘₯ ) ≀ 𝛼 } is a closed set. Equivalently, if the epigraph defined by epi 𝑓 = { ( π‘₯ , 𝑑 ) ∈ 𝑅 𝑛 + 1 | π‘₯ ∈ dom 𝑓 , 𝑓 ( π‘₯ ) ≀ 𝑑 } is closed, then the function 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.[1] This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it.

Closed convex function - Wikipedia Jump to content From Wikipedia, the free encyclopedia Terms in Maths In mathematics , a function f : R n β†’ R {\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} } is said to be closed if for each Ξ± ∈ R {\displaystyle \alpha \in \mathbb {R} } , the sublevel set { x ∈ dom f | f ( x ) ≀ Ξ± } {\displaystyle \{x\in {\mbox{dom}}f\vert f(x)\leq \alpha \}} is a closed set . Equivalently, if the epigraph defined by epi f = { ( x , t ) ∈ R n + 1 | x ∈ dom f , f ( x ) ≀ t } {\displaystyle {\mbox{epi}}f=\{(x,t)\in \mathbb {R} ^{n+1}\vert x\in {\mbox{dom}}f,\;f(x)\leq

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