Increasing

Increasing describes a change in which something becomes larger in amount, size, degree, or number over time or relative to a baseline. In everyday language it is used to describe rises in prices, temperatures, populations, or demand. The term is closely related to concepts such as growth, rise, and upward trend.

In mathematics, increasing refers to a relation that preserves order on an interval. A function f is increasing on an interval if, whenever x1 < x2, we have f(x1) ≤ f(x2). If the inequality is always strict, the function is strictly increasing. A sequence (a_n) is increasing if a_n ≤ a_{n+1} for all n. The opposite notions are decreasing (or strictly decreasing) and non-decreasing.

Common examples illustrate the idea. The exponential function f(x) = e^x is increasing on the real numbers, as is f(x) = x^3. The function f(x) = log x is increasing on its domain (0, ∞). By contrast, f(x) = x^2 is increasing only on [0, ∞) and not on all of R. In calculus, if the derivative f′ is positive on an interval, then f is strictly increasing there; if f′ is nonnegative, f is non-decreasing.

Applications and related concepts. Recognizing increasing behavior aids in modeling, forecasting, and analysis of trends in data. Related terms include monotone, non-decreasing, and growth rate. Understanding increasing functions and sequences provides a foundation for broader studies in analysis and applied disciplines.