monotonisuus

Monotoniisuus, eller monotonisuus, is mathematical term describing how a relation or function respects a given order. A function f defined on an ordered set is monotone if it preserves the order: on the real numbers, f is monotone increasing (nondecreasing) when x ≤ y implies f(x) ≤ f(y) for all x and y in the domain; it is monotone decreasing (nonincreasing) when x ≤ y implies f(x) ≥ f(y). If the implication is strict whenever x < y, f is strictly monotone, meaning f(x) < f(y) for x < y.

On an interval, monotone functions exhibit a simple directional behavior: a monotone increasing function never decreases as its input grows, and a monotone decreasing function never increases. They can have discontinuities, but only of the jump type, and they possess left- and right-hand limits at every point. The set of discontinuities of a real-valued monotone function on an interval is at most countable.

Examples illustrate the concept. The function f(x) = x^3 is strictly increasing on R; f(x) = -x is strictly decreasing. The absolute value function, |x|, is not monotone on the whole real line, but it is monotone on (-∞, 0] (decreasing) and on [0, ∞) (increasing).

Monotone sequences are sequences that are either nondecreasing or nonincreasing. If a monotone sequence is bounded, it converges to a limit (the supremum or infimum within the sequence’s range). In order theory, monotone maps are functions between partially ordered sets that preserve the order: x ≤ y implies f(x) ≤ f(y).

Monotonicity is a fundamental concept in analysis, topology, optimization, and computer science, where preserving order often ensures stability, convergence, and predictable behavior.