Hypograph (mathematics) explained

f:\Rⁿ → \R

is the set of points lying on or below its graph. A related definition is that of such a function's epigraph, which is the set of points on or above the function's graph.

The domain (rather than the codomain) of the function is not particularly important for this definition; it can be an arbitrary set instead of

Rⁿ

Definition

The definition of the hypograph was inspired by that of the graph of a function, where the of

f:X\toY

is defined to be the set

\operatorname{graph}f:=\left\{(x,y)\inX x Y~:~y=f(x)\right\}.

The or of a function

f:X\to[-infty,infty]

valued in the extended real numbers

[-infty,infty]=R\cup\{\pminfty\}

is the set

\begin{alignat}{4} \operatorname{hyp}f&=\left\{(x,r)\inX x R~:~r\leqf(x)\right\}\\ &=\left[f^-1(infty) x R\right]\cup

cup
	x\inf^-1(R)

(\{x\} x (-infty,f(x)]). \end{alignat}

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

\begin{alignat}{4} \operatorname{hyp}_Sf&=\left\{(x,r)\inX x R~:~r<f(x)\right\}\\ &=\operatorname{hyp}f\setminus\operatorname{graph}f\\ &=cup_x(\{x\} x (-infty,f(x))). \end{alignat}

Despite the fact that

might take one (or both) of

\pminfty

as a value (in which case its graph would be a subset of

X x R

), the hypograph of

is nevertheless defined to be a subset of

X x R

rather than of

X x [-infty,infty].

Properties

The hypograph of a function

is empty if and only if

is identically equal to negative infinity.

g:Rⁿ\toR

is a halfspace in

Rⁿ⁺¹.

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