Knowlpedia

Domain, epigraph, and proper function

dom(f) is where f is finite; epi(f) is the set above the graph; proper means dom(f)≠∅

Let XX be a vector space and let f:X→(−∞,+∞]f:X\to(-\infty,+\infty] be an function.

  • The domain of ff is
dom(f):={x∈X:f(x)<∞}.\mathrm{dom}(f):=\{x\in X: f(x)<\infty\}.
  • The epigraph of ff is
epi(f):={(x,α)∈X×R:f(x)≤α}.\mathrm{epi}(f):=\{(x,\alpha)\in X\times\mathbb{R}: f(x)\le \alpha\}.

The function ff is proper if dom⁡(f)≠∅\operatorname{dom}(f)\neq\varnothing. With the stated codomain, this is equivalent to the usual requirement that ff is never −∞-\infty and is not identically +∞+\infty.

Interpretation

The epigraph turns properties of ff into geometric properties of sets; for example, ff is exactly when its epigraph is convex.