Monotonic Function

(redirected from Nondecreasing function)

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 ?
xi] is a nondecreasing function of bounded variation with real values and satisfies [[integral].
j] is a nondecreasing function of the number of players selecting the same strategy.
In the following lemma he gives a characterization for [alpha]-SP weight for a nondecreasing function g on time scales.
H3) g(r) is a nondecreasing function on [0,1] with 0 = g(0) < g(1) < 1.
rho]] then [rho](ax) is a nondecreasing function of a [greater than or equal to] 0.
An Orlicz function M is a function M: [0, [infinity]) [right arrow] [0, [infinity]), which is continuous, convex, nondecreasing function define for x > 0 such that M(0) = 0, M(x) > 0 and M(x) [right arrow] [infinity] as x [right arrow] [infinity].
The concept of monotone operator may be viewed as a generalization of that of nondecreasing function on the real line.
Further, if f is the identity map on X, and [phi] is a continuous and nondecreasing function in Definition 1.
An Orlicz function M is a function M : [0,[infinity]) [right arrow] [0,[infinity]), which is continuous, convex, nondecreasing function define for x > 0 such that M(0) = 0, M(x) > 0 and M (x) [right arrow] [infinity] as x [right arrow] [infinity].
21) The analysis for any nondecreasing function can by treated by constructing a sequence of step functions that converges to b as K [right arrow] [infinity].
Let g(u) be a continuous nondecreasing function defined on [R.

Site: Follow: Share:
Open / Close