nonconstant

Nonconstant is a term used in mathematics to describe a relation, function, or sequence that is not fixed or uniform. Specifically, a function f defined on a domain D is nonconstant if there exist inputs x and y in D such that f(x) ≠ f(y). By contrast, a constant function assigns the same value to every input.

In analysis, the distinction between constant and nonconstant has several consequences. If a function is continuous on an interval and nonconstant, its range is an interval (the intermediate value property). A differentiable function that is nonconstant on an interval cannot have a derivative identically equal to zero; a function with derivative zero everywhere on an interval must be constant.

Examples help illustrate the idea. Nonconstant functions include f(x) = x, f(x) = x^2 on the real numbers, and f: R → R with f(x) = sin x. A constant function, such as f(x) = 7 for all x, is the opposite. In algebra, a polynomial is nonconstant if its degree is at least one, and such polynomials grow without bound as x grows in magnitude.

Nonconstant also applies to sequences. A sequence (a_n) is nonconstant if it takes at least two distinct values. Examples include a_n = (-1)^n or a_n = n. The concept underlies much of analysis and algebra, where distinguishing constant from nonconstant objects helps describe variability and potential for growth or change across a domain.