nichtmonotone

Nichtmonotone is a term used in mathematics to describe objects that do not have a monotone behavior with respect to a given order. In analysis and related fields, a function is called monotone if it preserves order: it is either nondecreasing or nonincreasing on its domain. Nichtmonotone therefore denotes the opposite of monotone.

For a real-valued function f defined on an interval, nichtmonotone means that there is no single direction of change consistent across the domain. Equivalently, f is not monotone if there exist points x1 < x2 < x3 with f(x1) ≤ f(x2) and f(x2) ≥ f(x3), or the reverse, showing both increasing and decreasing behavior somewhere on the interval.

Examples illustrate the idea. The function f(x) = x^2 on the real line is not monotone, since it decreases on (-∞, 0] and increases on [0, ∞). The trigonometric function f(x) = sin x on R is also nichtmonotone, displaying alternating rises and falls over any unbounded interval. A simple absolute value function, f(x) = |x|, is not monotone on the entire real line, though it is monotone on each of the half-lines [0, ∞) and (-∞, 0].

In discrete settings, nichtmonotone also describes Boolean functions that are not monotone with respect to the standard bitwise order. An example is the XOR function, where flipping an input bit from 0 to 1 can either increase or decrease the output. Nichtmonotone concepts appear in optimization and computer science, where nonmonotone behavior can complicate analysis and motivate methods that allow temporary deviations from a single monotone direction.