Epigraph (mathematics)
In mathematics, the epigraph or supergraph of a function f : X → [ − ∞ , ∞ ] {\displaystyle f:X\to [-\infty ,\infty ]} valued in the extended real numbers [ − ∞ , ∞ ] = R ∪ { ± ∞ } {\displaystyle [-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \}} is the set epi f = { ( x , r ) ∈ X × R : r ≥ f ( x ) } {\displaystyle \operatorname {epi} f=\{(x,r)\in X\times \mathbb {R} ~:~r\geq f(x)\}} consisting of all points in the Cartesian product X × R {\displaystyle X\times \mathbb {R} } lying on or above the function's graph. Similarly, the strict epigraph epi S f {\displaystyle \operatorname {epi} _{S}f} is the set of points in X × R {\displaystyle X\times \mathbb {R} } lying strictly above its graph.