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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.
As explained in Riesz & Sz.-Nagy (1990), every non-decreasing non-negative function F can be decomposed uniquely as a sum of a jump function f and a continuous monotone function g: the jump function f is constructed by using the jump data of the original monotone function F and it is easy to check that g = F − f is continuous and monotone. [10]
In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the good convergence behaviour of monotonic sequences, i.e. sequences that are non- increasing, or non- decreasing. In its simplest form, it says that a non-decreasing bounded -above sequence of real numbers converges to ...
A generalized quantifier GQ is said to be monotone increasing (also called upward entailing) if, for every pair of sets X and Y, the following holds: if , then GQ(X) entails GQ(Y). The GQ every boy is monotone increasing. For example, the set of things that run fast is a subset of the set of things that run.
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 ...
Intermediate value theorem: Let be a continuous function defined on [,] and let be a number with () < < ().Then there exists some between and such that () =.. In mathematical analysis, the intermediate value theorem states that if is a continuous function whose domain contains the interval [a, b], then it takes on any given value between () and () at some point within the interval.
Cyclical monotonicity. In mathematics, cyclical monotonicity is a generalization of the notion of monotonicity to the case of vector-valued function. [1][2]
In statistics, the monotone likelihood ratio property is a property of the ratio of two probability density functions (PDFs). Formally, distributions and bear the property if. that is, if the ratio is nondecreasing in the argument . If the functions are first-differentiable, the property may sometimes be stated.