In mathematics, the hypograph or subgraph of a function f : Rn → R is the set of points lying on or below its graph:
and the strict hypograph of the function is:
The set is empty if
The domain (rather than the co-domain) of the function is not particularly important for this definition; it can be an arbitrary set instead of
Similarly, the set of points on or above the function's graph is its epigraph.
Properties
A function is concave if and only if its hypograph is a convex set. The hypograph of a real affine function g : Rn → R is a halfspace in Rn+1.
A function is upper semicontinuous if and only if its hypograph is closed.