monotonic

Monotonic, in mathematics, describes a relation that preserves order.

A function f defined on an interval I is monotonic if it is either nondecreasing or nonincreasing. Specifically, f is monotone increasing (or nondecreasing) if x ≤ y implies f(x) ≤ f(y) for all x,y in I. It is monotone decreasing (or nonincreasing) if x ≤ y implies f(x) ≥ f(y). If the inequalities are strict for x < y, the function is strictly monotone (strictly increasing or strictly decreasing).

Examples include f(x) = x, which is strictly increasing on R, and f(x) = -x, which is strictly decreasing. The function f(x) = x^2 is not monotone on all of R, but it is monotone on [0, ∞) (increasing) and on (-∞, 0] (decreasing).

Properties and consequences: Monotone functions preserve order, meaning they respect the underlying arrangement of the domain. On an interval, a monotone function has one‑sided limits at every point, and a monotone function that is continuous has no sudden jumps. The set of discontinuities of a monotone function on an interval is at most countable, and they are all jump discontinuities.

Monotonicity extends to sequences: a sequence (a_n) is monotone if it is nondecreasing or nonincreasing. Every bounded monotone sequence converges, approaching its supremum or infimum.

In analysis and related fields, monotonicity underpins results such as the monotone convergence property and various order-theoretic constructions. It provides a simple yet powerful way to characterize and compare functions and sequences.