Decreasing

Decreasing is a term used to describe a downward movement in values or a tendency to become smaller as another variable increases. In everyday language it can refer to trends, prices, temperatures, or quantities that decline over time.

In mathematics, a real function f defined on an interval I is decreasing if x1 < x2 implies f(x1) ≥ f(x2). If the inequality is strict for all such pairs, the function is strictly decreasing. For a sequence a_n, the sequence is decreasing if n1 < n2 implies a_{n1} ≥ a_{n2}, with strict decrease requiring a_{n1} > a_{n2} for all pairs. A constant function or constant sequence is nonincreasing but not strictly decreasing.

Differentiable functions indicate a similar criterion: if f is differentiable on an interval and f'(x) ≤ 0 for all x in that interval, then f is decreasing there; if f'(x) < 0 everywhere, f is strictly decreasing.

Common examples include f(x) = -x, which decreases on the real line, and f(x) = e^{-x}, which decreases as x increases. The sequence a_n = 1/n is decreasing and converges to 0.

Decreasing trends are used in modeling decay processes, depreciation, cooling, inventories, and risk assessments. The term is often contrasted with increasing, and with monotone notions such as nonincreasing and strictly decreasing to distinguish weak versus strict declines.