Monotonic Function

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monotonic function

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McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Monotonic Function


(or monotone function), a function whose increments Δf(x) = f(x′) − f(x) do not change sign when Δx = x′ − x > 0; that is, the increments are either always nonnegative or always nonpositive. Somewhat inaccurately, a monotonic function can be defined as a function that always varies in the same direction. Different types of monotonic functions are represented in Figure 1. For example, the function y = x3 is an increasing function. If a function f(x) has a derivative f′(x) that is nonnegative at every point and that vanishes only at a finite number of individual points, then f(x) is an increasing function. Similarly, if f′(x) ≤ 0 and vanishes only at a finite number of points, then f(x) is a decreasing function.

Figure 1

A monotonicity condition can hold either for all x or for x on a given interval. In the latter case, the function is said to be monotonic on this interval. For example, the function y = Monotonic Function increases on the interval [−1,0] and decreases on the interval [0, +1]. A monotonic function is one of the simplest classes of functions and is continually encountered in mathematical analysis and the theory of functions. If f(x) is a monotonic function, then the following limits exist for any X0:


The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
Schlattmann, "An analytical approach to evaluating monotonic functions of fuzzy numbers," in Proceedings of the 8th Conference of the European Society for Fuzzy Logic and Technology, pp.
In summary, [OMEGA] is a monotonic function in the search segments; thus, Assumption 1 is verified.
(VI) The steps above are repeated until the final residual component r(x) becomes a monotonic function. The final estimation of the residue [??](x) is then considered.
Let C denote the set of completely monotonic functions. Some related references and a detailed collection of the most important properties of the completely monotonic functions can be found in [5] and [6, Chapter IV].
He graduated from Ferdinand I High School in 1944, completed his studies at the University of Bucharest's Faculty of Science, Department of Mathematics, in 1949, and obtained his PhD in Mathematics in 1956, with a thesis on the Monotonic functions of two variables, written under the direction of Miron Nicolescu.
Berg, "Some classes of completely monotonic functions," Annales Academiae Scientiarum Fennicae Mathematica, vol.
Furthermore, our method is more convenient in computation than the convex relaxation [19] because the main work is to solve the linear programs and the zeros of strictly monotonic functions of one variable over the interval [0,1), which can be solved very efficiently by the existing methods, for example, by the simplex method and the bisection search method.
Merovci, Logarithmically completely monotonic functions involving the generalized Gamma Function, Le Matematiche LXV, Fasc., II (2010), 15-23.
They discuss such topics as K-trivials never being continuously random, limitwise monotonic functions and their applications, a dichotomy for the Machey Borel structure, automatic families, and computable Dowd-type generic oracles.
Completely monotonic functions involving ln[GAMMA] (x) are important because they produce sharp bounds for the polygamma functions, see, e.g., [2, 4, 17, 20-29].
We will take care of two types of functions: one is the family of monotonic functions with singularities, which normally are computed as 'improper' Riemann integrals, but now they are direct results from Henstock-Kurzweil definition.

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