monotonicitet

Monotonicitet, or monotonicity, is a property of a function or sequence with respect to an underlying order. In mathematics, a function f defined on an ordered set is called monotone if it preserves the order: for any x and y in the domain, x ≤ y implies f(x) ≤ f(y). Monotonicity can be non-strict (nondecreasing or nonincreasing) or strict (increasing or decreasing).

Two common variants are monotone increasing and monotone decreasing. An increasing function satisfies x ≤ y implies f(x) ≤ f(y); if the inequality is strict for all x < y, the function is strictly increasing. Decreasing is defined analogously with reversed inequality. Many authors treat nondecreasing and nonincreasing as the same as monotone increasing/decreasing, while others maintain the distinction between strict and non-strict forms.

In sequences, monotonicity is defined similarly: a sequence (a_n) is monotone increasing if a_n ≤ a_{n+1} for all n, and monotone decreasing if a_n ≥ a_{n+1}. Strict variants require strict inequalities. Monotone sequences that are bounded converge; this is the monotone convergence theorem in real analysis.

Properties: On an interval, differentiable functions with nonnegative derivative are nondecreasing; with positive derivative on a subinterval, the function is strictly increasing there (under suitable regularity). Monotone functions on real intervals have at most countably many discontinuities and possess left- and right-hand limits at every point.

In order theory, monotone (order-preserving) maps between partially ordered sets satisfy x ≤ y implies f(x) ≤ f(y). These maps preserve and reflect ordered structure and are central in lattice theory and related areas.