Differentierbar

Differentierbar, often rendered in German as differenzierbar, is a mathematical term describing when a function has a derivative at a point or on a domain. In real analysis, a function f: U → R is differentiable at a point a ∈ U if there exists a linear map Df(a) such that f(a+h) = f(a) + Df(a)h + o(||h||) as h → 0. In one-variable terms, this means the usual derivative f′(a) exists.

Differentiability implies continuity, but the converse is not true. For example, f(x) = |x| is continuous on R but not differentiable at x = 0. In higher dimensions, a function f: U ⊆ R^n → R^m is differentiable at a if there is a linear map Df(a): R^n → R^m with f(a+h) = f(a) + Df(a)h + o(||h||). The matrix Df(a) is the Jacobian, encoding all first-order partial derivatives.

Differentiability connects to several broader concepts. The chain rule describes how derivatives compose under differentiable mappings. Classes of differentiability include C^1 (once differentiable with a continuous derivative), C^k (k times differentiable), and C^∞ (infinitely differentiable). Real-analytic functions are those locally equal to their power series.

In geometry and analysis, differentiable maps between manifolds are the foundation of smooth structures, enabling calculus on curved spaces. While many elementary functions (polynomials, exponentials, logarithms on their domains) are differentiable everywhere in their domains, piecewise definitions may fail to be differentiable at joining points.