Monotonic Function

(redirected from Nondecreasing function)

monotonic function

[¦män·ə¦tän·ik ′fəŋk·shən]

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 ?
rho]] then [rho](ax) is a nondecreasing function of a [greater than or equal to] 0.
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.
Clearly n(x, y) is nonnegative and nondecreasing function for x, y [member of] [R.
1 from (Stoica & Megan 2008), there exists a nondecreasing function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
optimal output is a nondecreasing function of the firm's index (Arrow-Pratt measure) of risk aversion.
n], suitably rescaled, converge pointwise to a continuous, nondecreasing function s : [0,1] [right arrow] [0,1].
This method constrains the particulate air pollution mortality concentration-response function to be biologically plausible--that is, a nondecreasing function of the particulate air pollution concentration.