zeroderivative

Zeroderivative is a term used in mathematics to describe a function whose derivative is identically zero on its domain. In other words, if a function f is differentiable on a set and f'(x) = 0 for every point in that set, then f is said to have a zeroderivative on that domain.

A fundamental consequence is that on any interval where the function is differentiable and its derivative is zero everywhere, the function must be constant on that interval. This follows from the mean value theorem: for any two points a and b in the interval, f(b) − f(a) = f'(c)(b − a) for some c between a and b, and since f'(c) = 0, we obtain f(b) = f(a). Therefore, the zeroderivative condition implies constancy on connected domains. In higher dimensions, a differentiable function f: U ⊂ R^n → R with gradient zero on a connected open set U is likewise constant on U.

The concept is not universal terminology; in many contexts a function with zeroderivative is simply described as a constant function. A caveat is that derivative zero almost everywhere does not guarantee constancy. The Cantor function is continuous and differentiable almost everywhere with derivative zero on almost all points but is not constant, illustrating the distinction between zero derivative everywhere and zero derivative almost everywhere.

Examples of zeroderivative functions include any constant function, such as f(x) = c, or the zero function f(x) = 0. Related ideas include the derivative, gradient, the mean value theorem, and the Cantor function.