nichtderivative

Nichtderivative is a non-standard term used to refer to a function that is not the derivative of any other function on a given domain. In this sense, a nichtderivative is a function that cannot be realized as F′(x) for any function F whose domain includes the same interval.

Key idea and properties

Derivatives satisfy the Darboux property, meaning they have the intermediate value property: if a and b are points in the interval and f(a) and f(b) take two values, then f takes every value between them on the interval. This property implies that many intuitive “jumps” or discontinuities cannot occur in a derivative. Consequently, a function with a jump discontinuity or a function that clearly fails the intermediate value property cannot be a derivative, and thus would be a nichtderivative on that interval.

Examples

The Dirichlet function, which takes 1 on rational points and 0 on irrational points, does not have the intermediate value property and is not a derivative. Likewise, a step function with a genuine jump, such as f(x) = 0 for x < 0 and f(x) = 1 for x ≥ 0 on an interval, cannot be a derivative. By contrast, many smooth or piecewise smooth functions are derivatives of some antiderivative.

Context and usage

Nichtderivative is not a standard technical term in calculus, where the conventional notions are derivative, antiderivative, and primitive function. The concept is often discussed implicitly when characterizing which functions can occur as derivatives.

See also

Derivative, Antiderivative, Darboux property, Baire class 1.