# Monotonic Function

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## monotonic function

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(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:

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References in periodicals archive ?
([H.sub.0]) g(0, y) = 0, for ally [greater than or equal to] 0; ([partial derivative]g/[partial derivative]x)(x, y) [greater than or equal to] 0 (or g(x, y) is a strictly monotone increasing function with respect to x when f [equivalent to] 0) and ([partial derivative]g/[partial derivative]y)(x, y) [less than or equal to] 0, for all x [greater than or equal to] 0 and y [greater than or equal to] 0.
This VI formulation is strictly monotone and Lipschitz continuous, permitting a number of existing efficient algorithms for its solution [10, 26].
is strictly monotone, so by [29, Proposition 25.10], [[omega].sub.i] is strictly convex for i = 1, ..., n.
* if [alpha] and [beta] are almost surely strictly monotone functions on RanX and RanY, respectively, then:
for all x, y [member of] X; [J.sub.X,[phi]] is strictly monotone if and only if X is strictly convex.
where the continuous function a : [R.sub.+] [right arrow] [R.sub.+] is strictly monotone increasing with a(0) = 0 and a(t) [right arrow] [infinity] as t [right arrow] [infinity].
and [{-((1 + [square root of (1 + [x.sup.2] + x)])x/[(x+1).sup.2])}.sup.[infinity].sub.x=1] is strictly monotone increasing; we also have -1 + [square root of (3)]/4 = 0.1830127 ...
where [phi](t) is a differentiable function, [alpha](t) [member of] [C.sup.1] [0, T] is a strictly monotone increasing function and satisfies that -[tau] [less than or equal to] [alpha](t) [less than or equal to] t and [alpha](0) = -[tau], there exists [t.sub.1] [member of] [0, T] such that [alpha]([t.sub.1]) = 0, and f:D = [0,T] x R x R x R is a given continuous mapping and satisfies the Lipschitz condition
Dhage: Hybrid fixed point theory for strictly monotone increasing multivalued mappings with applications, Comput.
(1) [[phi].sub.1]: [0,c] [right arrow] [0, 1], strictly monotone increasingC'bijective map [[phi].sub.1], [[phi].sup.-1.sub.1] are absolutely continuous

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