Monotonic Function

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monotonic function

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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:


References in periodicals archive ?
That [tau](s) must be monotone decreasing to zero, as s [right arrow] [infinity], a and such that [tau](0) = 1.
I](I) is the set of all monotone decreasing intuitionistic Fuzzy set [Amember of][[zeta].
Cr] for t [member of] R, we define a monotone decreasing family {[V.
The monotone decreasing top segment exists if and only if [Q.
decreasing top segment, the monotone decreasing bottom segment, monotone increasing bottom segment, their sufficient and necessary conditions are
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.
form a monotone decreasing pattern so that we reach the minimum, necessary, after a finite number of iterations not exceeding n - 1.
N} x R [right arrow] R is a continuous function, monotone decreasing in the second and the third variables, and the existence of a pair of well ordered lower and upper solutions, that is, the lower solution is less than or equals to the upper one, P W.
2] is a monotone decreasing bijective mapping'C Then we can consider inverse mapping [[phi].
1] is monotone decreasing, if we note it bijective mapping.
When f'(x) is monotone decreasing in interval (a, b), the proof is same to above proof.