The definition of the hypograph was inspired by that of the graph of a function, where the graph of ${\displaystyle f:X\to Y}$  is defined to be the set

${\displaystyle \operatorname {graph} f:=\left\{(x,y)\in X\times Y~:~y=f(x)\right\}.}$ 

The hypograph or subgraph of a function ${\displaystyle f:X\to [-\infty ,\infty ]}$  valued in the extended real numbers ${\displaystyle [-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \}}$  is the set^[2]

${\displaystyle {\begin{alignedat}{4}\operatorname {hyp} f&=\left\{(x,r)\in X\times \mathbb {R} ~:~r\leq f(x)\right\}\\&=\left[f^{-1}(\infty )\times \mathbb {R} \right]\cup \bigcup _{x\in f^{-1}(\mathbb {R} )}(\{x\}\times (-\infty ,f(x)]).\end{alignedat}}}$ 

Similarly, the set of points on or above the function is its epigraph. The strict hypograph is the hypograph with the graph removed:

${\displaystyle {\begin{alignedat}{4}\operatorname {hyp} _{S}f&=\left\{(x,r)\in X\times \mathbb {R} ~:~r<f(x)\right\}\\&=\operatorname {hyp} f\setminus \operatorname {graph} f\\&=\bigcup _{x\in X}(\{x\}\times (-\infty ,f(x))).\end{alignedat}}}$ 

Despite the fact that ${\displaystyle f}$  might take one (or both) of ${\displaystyle \pm \infty }$  as a value (in which case its graph would not be a subset of ${\displaystyle X\times \mathbb {R} }$ ), the hypograph of ${\displaystyle f}$  is nevertheless defined to be a subset of ${\displaystyle X\times \mathbb {R} }$  rather than of ${\displaystyle X\times [-\infty ,\infty ].}$ 

The hypograph of a function ${\displaystyle f}$  is empty if and only if ${\displaystyle f}$  is identically equal to negative infinity.

A function is concave if and only if its hypograph is a convex set. The hypograph of a real affine function ${\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} }$  is a halfspace in ${\displaystyle \mathbb {R} ^{n+1}.}$ 

A function is upper semicontinuous if and only if its hypograph is closed.

• Effective domain
• Epigraph (mathematics) – the set of points lying on or above the graph of a functionPages displaying wikidata descriptions as a fallback
• Proper convex function

• Rockafellar, R. Tyrrell; Wets, Roger J.-B. (26 June 2009). Variational Analysis. Grundlehren der mathematischen Wissenschaften. Vol. 317. Berlin New York: Springer Science & Business Media. ISBN 9783642024313. OCLC 883392544.