Nonderivative

Nonderivative is a term sometimes used informally to describe a function that cannot be expressed as the derivative of any other function on a given interval. In precise terms, a function f defined on an interval is a nonderivative if there is no differentiable function F on that interval with F'(x) = f(x) for all x in the interval. The phrase is not standard in rigorous textbooks, where one more often says “not in the image of the differentiation operator.”

A key property of derivatives is the Darboux (intermediate value) property: if F is differentiable, then F' has the intermediate value property. Therefore a function that fails this property cannot be a derivative of any function on that interval. A classic simple example is the Heaviside step function, which has a jump discontinuity and thus does not possess the intermediate value property. More generally, any function with a jump or similar discontinuity cannot be a derivative. However, possessing the intermediate value property is not by itself sufficient to guarantee that a function is a derivative, and there exist functions with the intermediate value property that are not derivatives.

In practice, whether a given function is a nonderivative can depend on the domain and on the regularity requirements placed on the primitive function. The concept highlights that not every function arises as a derivative, even though many common functions (polynomials, exponentials, trigonometric functions, and many constructed derivatives) do appear as derivatives.

See also: derivative, differentiability, Darboux property, Darboux's theorem, image of the differentiation operator.