monotonically

Monotonically is the adverb used to describe a quantity that changes in a way that preserves order with respect to its argument or index. In mathematics, a function is described as monotone if it preserves order in a nondecreasing or nonincreasing fashion. Specifically, a function f is monotone increasing if x1 ≤ x2 implies f(x1) ≤ f(x2), and monotone decreasing if x1 ≤ x2 implies f(x1) ≥ f(x2). When the inequalities are strict for all distinct inputs, the function is often called strictly monotone. Some sources distinguish between monotone (nondecreasing or nonincreasing) and strictly monotone, while others use monotone and increasing/decreasing more loosely; the phrase monotonically increasing/monotone increasing is commonly used to emphasize nondecreasing behavior.

The term also applies to sequences and more general processes. A sequence {a_n} is monotonically increasing if a_n ≤ a_{n+1} for all n, and monotonically decreasing if a_n ≥ a_{n+1} for all n. The word can describe a function on an interval, a sequence, or a process that preserves order as its input or index grows. Notable examples include f(x) = x^3, which is monotonically increasing on the real line, and a sequence a_n = 1 − 1/n, which increases monotonically toward 1. Conversely, a_n = (−1)^n is not monotone.

In analysis, monotonicity yields useful consequences: a monotone sequence that is bounded converges, and monotone functions on closed intervals have well-behaved integrability properties. Monotonicity is a fundamental, widely used concept across calculus, real analysis, and related fields.