Hypograph (mathematics)

In mathematics, the hypograph or subgraph of a function

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[1] instead of

is defined to be the set The hypograph or subgraph of a function

valued in the extended real numbers

as a value (in which case its graph would not be a subset of

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

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

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