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Hypograph of a function. 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.
Epigraph of a function A function (in black) is convex if and only if the region above its graph (in green) is a convex set.This region is the function's epigraph. In mathematics, the epigraph or supergraph [1] of a function: [,] valued in the extended real numbers [,] = {} is the set = {(,) : ()} consisting of all points in the Cartesian product lying on or above the function's graph. [2]
Hypograph may refer to: Hypograph (mathematics) , the set of points lying below the graph of a function Hypograph, or hypogram, something written at the end of a document (for example, a postscript )
The epigraphs of extended real-valued functions play a role in convex analysis that is analogous to the role played by graphs of real-valued function in real analysis. Specifically, the epigraph of an extended real-valued function provides geometric intuition that can be used to help formula or prove conjectures.
In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity.An extended real-valued function is upper (respectively, lower) semicontinuous at a point if, roughly speaking, the function values for arguments near are not much higher (respectively, lower) than ().
Epigraph may refer to: An inscription, as studied in the archeological sub-discipline of epigraphy; Epigraph (literature), a phrase, quotation, or poem that is set at the beginning of a document or component; Epigraph (mathematics), the set of points lying on or above the graph of a function
What is typically used is y vs. x, such that x is horizontal and y is vertical. However, when specifically talking about plotting a function vs. its input, it is more clear and intuitive to plot f(x) vs. x (or f(y) vs. y or whatever), since the variables x and y are just placeholders. EmergencyBackupChicken 17:00, 7 May 2009 (UTC)
Equivalently, a concave function is any function for which the hypograph is convex. The class of concave functions is in a sense the opposite of the class of convex functions. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex.