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 of the function is not particularly important for this definition; it can be an arbitrary set instead of.
Definition
The definition of the hypograph was inspired by that of the graph of a function, where the of is defined to be the setThe or of a function valued in the extended real numbers is the set
Similarly, the set of points on or above the function is its epigraph.
The is the hypograph with the graph removed:
Despite the fact that might take one of as a value, the hypograph of is nevertheless defined to be a subset of rather than of
Properties
The hypograph of a function is empty if and only if 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 is a halfspace in
A function is upper semicontinuous if and only if its hypograph is closed.