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 instead of . The definition of the hypograph was inspired by that of the graph of a function, where the of is defined to be the set The 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 (or both) of as a value (in which case its graph would be a subset of ), the hypograph of is nevertheless defined to be a subset of rather than of 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.