Monotonic Function

(redirected from Monotonically decreasing)

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 = 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:

and

References in periodicals archive ?
n,k], 1 [less than or equal to] k [less than or equal to] n, and it is a monotonically decreasing function in the open intervals (-[infinity], [x.
monotonically decreasing relationship between both.
The unique equilibrium strategy function takes the form of a monotonically decreasing time [T.
The absolute values of the own-price elasticities for each food category are monotonically decreasing from the lowest quintile to the top quintile income groups.
is monotonically decreasing for (ln n)/n < p < 1 - (ln n)/n as n [right arrow] [infinity], where [psi](1) is given by (6) and r is defined in (4).
In the first few minutes of shearing, the stress reached a maximum before monotonically decreasing, as shown in Fig.
if everyone has monotonically decreasing marginal utility in income, and if these utility functions are sufficiently generic to justify the assumption of a representative agent, then high-income individuals will be farther down the marginal utility function than low-income individuals.
Parts (ii) and (iii) of the proposition state the interesting result that a pure probationary period will be chosen whenever the relative risk ratio between high and low is monotonically decreasing, and that a pure limited term will be used in the opposite case.
k,l]) be a monotonically decreasing double sequence of nonnegative numbers.
If f is a positive r-concave function, then G(t,q) is monotonically decreasing on [0,1] .