Monotonicity

Monotonicity is a property of functions and sequences describing a consistent directional behavior with respect to the order of their inputs. A function f defined on an interval I is monotone increasing (nondecreasing) if x1 < x2 implies f(x1) ≤ f(x2) for all x1, x2 in I. It is monotone decreasing (nonincreasing) if x1 < x2 implies f(x1) ≥ f(x2). Strict monotonicity uses strict inequalities throughout, so x1 < x2 implies f(x1) < f(x2) for increasing, or f(x1) > f(x2) for decreasing. Monotonicity is often defined on domains with a total order, such as intervals of the real line.

A sequence (a_n) is monotone increasing if a_{n+1} ≥ a_n for all n, and monotone decreasing if a_{n+1} ≤ a_n. Monotone sequences have well-defined limits in the extended real numbers; an increasing sequence bounded above converges to its supremum, while a decreasing sequence bounded below converges to its infimum.

Key properties: On any interval, a monotone function has limits from the left and right at every point, and it has at most countably many discontinuities, all of which are jump discontinuities. Monotone functions are of bounded variation. If a function is differentiable and its derivative satisfies f′(x) ≥ 0 (or ≤ 0) on an interval, then f is monotone increasing (or decreasing) on that interval; the converse is not generally true.

Examples include f(x) = x^3 (increasing), f(x) = -x (decreasing), and f(x) = ⌊x⌋ (nondecreasing but with jumps). Monotonicity concepts are fundamental in analysis and underpin results such as the monotone convergence theorem and order-preserving maps in ordered structures.