# 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 = 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:

and

References in periodicals archive ?
1] > 0, [OMEGA] is a monotonic function within the range of u = [0,1] and [[OMEGA].
The simplified modeling script of only one biologic component involving monotonic functions of efficiency is not always verified in experimental surveys.
Let L on (0, [infinity]) stand for the set of logarithmically completely monotonic functions.
Chen, Four logarithmically completely monotonic functions involving gamma function, J.
These results can be extended to the case where V consists of a sum of monotonic functions [[phi].
When gathering demographic data, one is not surprised by occasional deviations from smooth or monotonic functions.
Because W(t) and R(t) are monotonic functions, there is no need for second-order conditions, as we can be assured that the t that makes L* = 0 will, indeed, be at a minimum of L.
The text covers the Stieltjes integral, fundamental formulas, the moment problem, absolutely and completely monotonic functions, Tauberian theorems, the bilateral Laplace transform, inversion and representation problems for the Laplace transform, and the Stieltjes transform.
The main properties of completely monotonic functions are given in [3, Chapter IV].
Then the theory of monotonic functions [2, 13; 7, 197] is applied to study the effect of a global change from a constant price contract to a price-quality schedule contract.

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