Domain, epigraph, and proper function

Let $X$ be a vector space and let $f:X\to \mathbb{R}$ be an extended-real-valued function (see extended reals ).

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

The function $f$ is proper if $\mathrm{dom}(f)\neq\emptyset$.

Context. The epigraph turns function properties into geometric properties of sets; convexity of $f$ is defined by convexity of $\mathrm{epi}(f)$ (see convex functions via epigraphs ).