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A function that is not monotonic. In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. [ 1 ][ 2 ][ 3 ] This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.
Gradients of convex functions are cyclically monotone. In fact, the converse is true. [ 4 ] Suppose U {\displaystyle U} is convex and f : U ⇉ R n {\displaystyle f:U\rightrightarrows \mathbb {R} ^{n}} is a correspondence with nonempty values.
hide. In mathematics, Helly's selection theorem (also called the Helly selection principle) states that a uniformly bounded sequence of monotone real functions admits a convergent subsequence. In other words, it is a sequential compactness theorem for the space of uniformly bounded monotone functions. It is named for the Austrian mathematician ...
Discontinuities of monotone functions. In the mathematical field of analysis, a well-known theorem describes the set of discontinuities of a monotone real-valued function of a real variable; all discontinuities of such a (monotone) function are necessarily jump discontinuities and there are at most countably many of them.
Dini's theorem. In the mathematical field of analysis, Dini's theorem says that if a monotone sequence of continuous functions converges pointwise on a compact space and if the limit function is also continuous, then the convergence is uniform. [1]
Thomae's function. Point plot on the interval (0,1). The topmost point in the middle shows f (1/2) = 1/2. Thomae's function is a real -valued function of a real variable that can be defined as: [1]: 531. It is named after Carl Johannes Thomae, but has many other names: the popcorn function, the raindrop function, the countable cloud function ...
Print/export Download as PDF; Printable version; From Wikipedia, the free encyclopedia. Redirect page. Redirect to: Monotonic function; Retrieved from " ...
Operator monotone function. In linear algebra, the operator monotone function is an important type of real-valued function, fully classified by Charles Löwner in 1934. [1] It is closely allied to the operator concave and operator concave functions, and is encountered in operator theory and in matrix theory, and led to the Löwner–Heinz ...