ableitbare

Ableitbare is a term used in German mathematics to describe a function or mapping that has a derivative at points of its domain, i.e., that is differentiable. The adjective derives from ableiten (to differentiate) with the suffix -bar indicating capability, so ableitbar means "capable of being differentiated." In practice, one speaks of a "ableitbare Funktion" or "eine Funktion, die ableitbar ist," which in English is a differentiable function.

Definition and scope: A real-valued function f defined on an interval I is ableitbar on I if for every x in I the derivative f'(x) exists, equivalently if the limit of (f(x+h)−f(x))/h as h approaches zero exists. If this holds for all points in I, f is differentiable on I. Differentiability implies continuity, but the converse is not always true; for example, the absolute value function is continuous everywhere but not differentiable at x = 0.

Higher-order and related concepts: If f is ableitbar on I and f' is itself differentiable, then f is twice differentiable on I, and so on. The classification C^k describes functions with continuous derivatives up to order k, while C^∞ denotes smooth functions. In several variables or in functional analysis, notions such as Fréchet differentiability generalize the idea of ableitbare to mappings between normed spaces or Banach spaces.

Examples and non-examples: Polynomials, exponential, logarithmic, and trigonometric functions are ableitbar on their domains, typically the whole real line. The function |x| is not ableitbar at x = 0. In higher dimensions, a function can be differentiable at a point even if it is not differentiable elsewhere.

See also: derivative, differentiability, calculus, Fréchet differentiability.

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