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In mathematics, the epigraph or supergraph of a function f : X → {\displaystyle f:X\to } valued in the extended real numbers = R ∪ { ± ∞ } {\displaystyle =\mathbb {R} \cup \{\pm \infty \}} is the set, denoted by epi f , {\displaystyle \operatorname {epi} f,} of all points in the Cartesian product X × R {\displaystyle X\times \mathbb {R} } lying on or above its graph. 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.
Importantly, although both the graph and epigraph of f {\displaystyle f} consists of points in X × , {\displaystyle X\times ,} the epigraph consists entirely of points in the subset X × R , {\displaystyle X\times \mathbb {R} ,} which is not necessarily true of the graph of f . {\displaystyle f.} If the function takes ± ∞ {\displaystyle \pm \infty } as a value then graph f {\displaystyle \operatorname {graph} f} will not be a subset of its epigraph epi f . {\displaystyle \operatorname {epi} f.} For example, if f = ∞ {\displaystyle f\left=\infty } then the point ] = {\displaystyle \left\right]=\left} will belong to graph f {\displaystyle \operatorname {graph} f} but not to epi f . {\displaystyle \operatorname {epi} f.} These two sets are nevertheless closely related because the graph can always be reconstructed from the epigraph, and vice versa.
The study of continuous real-valued functions in real analysis has traditionally been closely associated with the study of their graphs, which are sets that provide geometric information about these functions. Epigraphs serve this same purpose in the fields of convex analysis and variational analysis, in which the primary focus is on convex functions valued in {\displaystyle } instead of continuous functions valued in a vector space. This is because in general, for such functions, geometric intuition is more readily obtained from a function's epigraph than from its graph. Similarly to how graphs are used in real analysis, the epigraph can often be used to give geometrical interpretations of a convex function's properties, to help formulate or prove hypotheses, or to aid in constructing counterexamples.