différentiable

Différentiable, in mathematics, refers to the property of a function having a linear approximation at a point. Formally, a function f: U ⊆ R^n → R^m is differentiable at x0 ∈ U if there exists a linear map Df(x0): R^n → R^m such that f(x0 + h) = f(x0) + Df(x0)h + o(‖h‖) as h → 0. The map Df(x0) is called the derivative at x0 (often represented by the Jacobian matrix when m and n are finite). In the one-variable case, Df(x0) corresponds to the usual derivative f′(x0).

If a function is differentiable at a point, it is continuous there. Differentiability does not necessarily require the function to have all partial derivatives, but a common sufficient condition is that all partial derivatives exist in a neighborhood of x0 and are continuous at x0; then f is differentiable at x0 (C^1 near x0).

Differentiability can be extended to higher orders. A function is of class C^k if its derivatives up to order k exist and are continuous; C^∞ means derivatives of every order are continuous; real-analytic means locally equal to a convergent power series. In multivariable analysis, the derivative Df(x0) is represented by the Jacobian matrix; higher-order derivatives require multilinear maps such as Hessians.

Examples: f(x)=x^2 is differentiable everywhere with f′(x)=2x. f(x)=|x| fails to be differentiable at x=0. In several variables, f(x,y)=x^2+y^2 is differentiable everywhere, with Jacobian [2x, 2y].

Some functions are differentiable almost everywhere but not everywhere (e.g., Lipschitz functions on R^n, by Rademacher's theorem). Differentiability is stronger than continuity and is the key to defining gradients and Jacobians used in optimization and analysis.