# 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 ?
It is easy to see that for J' [[subset].bar] J, [THETA](J) [[subset].bar] [THETA](J) and the same for [PHI]; thus [THETA] and [PHI] are order-preserving bijections and therefore preserve all meets and joins.
For example, user has no permission to create UDFs in the cloud database, and CryptDB uses mOPE (mutable order-preserving encoding) to support order operations.
To date, many fully homomorphic encryption (FHE) and order-preserving encryption (OPE) schemes were proposed [5-10].
Nussbaum, A Collatz-Wielandt characterization of the spectral radius of order-preserving homogeneous maps on cones, 2011, https://arxiv.org/abs/1112.5968.
g : [0, [infinity]) x [R.sup.n] [right arrow] [R.sup.n] is order-preserving on [R.sup.n.sub.+] if g(t, x) [greater than or equal to] g(t, y) for any t [member of] [0, [infinity]) and any x, y [member of] [R.sup.n.sub.+] satisfying x [greater than or equal to] y.
In particular, we refer to a classical theorem of White [20], according to which every maximal element for a preorder is determined by maximizing an order-preserving function (provided that an order-preserving function exists).
A vector field g : [R.sup.n] [right arrow][R.sup.n] is said to be order-preserving on [R.sup.n.sub.+] \ {0}, if g(x) [greater than or equal to] g(y) for any x, y [member of] [R.sup.n.sub.+] \ {0} such that x [greater than or equal to] y.
Hess, "Stability of fixed points for order-preserving discrete-time dynamical systems," Journal fur die Reine und Angewandte Mathematik, vol.
For symmetric setting, the authors in [4, 13, 14] introduced ranked keyword search schemes based on order-preserving encryption technique or two-round protocol, which allows the server to only return the top-k relevant results to the user.
This is also an EL-labeling since the map [delta] [??] [[delta].sup.-1] is order-preserving and [sigma] = [pi] x (i,j) if and only if [[sigma].sup.-1] = (i,j) x [[pi].sup.-1].

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