Monotonic Function


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monotonic function

[¦män·ə¦tän·ik ′fəŋk·shən]
(mathematics)

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:

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References in periodicals archive ?
It is very similar to the MP-BS-OWAWA operator with the difference that we add a strictly continuous monotonic function for the WA and the OWA that includes a wide range of particular cases including quadratic and geometric aggregations.
As claimed earlier, the statistical yield predicted by this model is a monotonic function of the variance of the underlying random process, in accord with the modern view of statistical psychokinesis.
As demonstrated for various types of binary blends, the blend modulus is always a monotonic function of the blend composition because no interfacial debonding occurs at small strains, at which the blend moduli are routinely measured.
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].
Merovci, Logarithmically completely monotonic functions involving the generalized Gamma Function, Le Matematiche LXV, Fasc.
Grey prediction requires that original time series should be non-negative monotonic functions and can accord with exponential laws.
5 since [upsilon]([theta]) and [eta]([theta]) are not monotonic functions on [-1, 0].
Completely monotonic functions associated with the gamma function and its q-analogues.
Modeling reading rate performance could be constructed as the difference of two monotonic functions and be the algebraic sum of increase and decrease in reading rate with print size increase.

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