# 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 ?
We strictly define t as the monotone increasing function of both [n.
With that in mind, we will show that a unique monotone increasing, bounded equilibrium bid function exists in the case when W = 3 and costs are uniformly distributed.
d,l](s) is monotone increasing in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
is strictly monotone increasing with a(0) = 0 and a(t) [right arrow] [infinity] as t [right arrow] [infinity].
MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] For a given exponent [alpha] [member of] (0,1/2), there exists a monotone increasing function l(*) such that
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q [member of] Q] be monotone increasing collections of an intuitionistic Fuzzy smooth quasi uniform decreasing open [F.
The monotone increasing top segment exists if and only if [Q.
GAMMA](s) exist the least value in open interval [1,2], and monotone decreasing at the left side of the point, at the right side of the point monotone increasing.
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For 3 [less than or equal to] k [less than or equal to] n - 1, it is clear that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a strictly monotone increasing function of t.

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