Nonconstancy

Nonconstancy refers to the property of not being constant. In mathematics and related fields, a function, sequence, or mapping is nonconstant if it does not assign the same value to every input. More formally, a function f: X → Y is nonconstant if there exist x1, x2 in X with f(x1) ≠ f(x2); equivalently, its image contains at least two distinct values.

In real analysis, a nonconstant real-valued function defined on an interval takes at least two different values. Nonconstancy is a common precondition for many theorems, such as those asserting the existence of points with varying behavior. It is important to note that a nonconstant function can exhibit substantial local flatness: a differentiable function on an interval may have a derivative that vanishes on large sets while still being nonconstant (the Cantor function is a classic example). Conversely, if a function is differentiable on an interval and its derivative is identically zero there, the function is constant on that interval.

In algebra, a nonconstant polynomial is one whose degree is at least one; constant polynomials have degree zero. Nonconstant polynomials define more complex algebraic and geometric objects, such as curves with multiple points where the polynomial vanishes.

In probability, a random variable is nonconstant if it does not take a single value with probability one; constant random variables have degenerate distributions.

Nonconstancy signals variability across a domain and contrasts with constancy, the property of uniform sameness. See also constancy, variability, derivative, polynomial, random variable.