monotoneincreasing

Monotone increasing, sometimes written monotone-increasing, is a term used in mathematics to describe a relationship that preserves order. For a real-valued function f defined on a set D, f is monotone increasing on D if x ≤ y implies f(x) ≤ f(y) for all x, y in D. If the implication is strict (x < y implies f(x) < f(y)), f is strictly increasing. Some authors use non-decreasing to mean monotone increasing.

The concept also applies to sequences. A sequence (a_n) is monotone increasing if a_n ≤ a_{n+1} for every n; it is strictly increasing if a_n < a_{n+1} for all n.

Examples: The function f(x) = x is monotone increasing on the real numbers; f(x) = e^x and f(x) = x^3 are increasing on R. The function f(x) = x^2 is not monotone increasing on all of R, but it is increasing on [0, ∞).

Extensions and related ideas: In order theory, a function between ordered sets is monotone (isotone) if it preserves order: x ≤ y implies f(x) ≤ f(y). Monotone functions are often studied for their basic properties, such as preserving limits of monotone sequences under certain conditions, and having bounded variation on intervals.

Notes: Monotonicity can simplify analysis and optimization, since monotone objects behave predictably with respect to order. Discontinuities of monotone functions on an interval are limited to at most countably many points.