Differensierbare

Differensierbare, or differentiable, describes a property of a function indicating that its rate of change can be described by a linear approximation at each point in its domain. In common calculus, a real-valued function of one variable is differentiable at a point if the derivative at that point exists, meaning the limit of the average rate of change as the increment tends to zero exists.

Formally, for a function f: R -> R, f is differentiable at x0 if the limit lim(h -> 0) [f(x0 + h) − f(x0)] / h exists. In several variables, differentiability is stronger: a function f: R^n -> R^m is Fréchet differentiable at x0 if there exists a linear map L such that lim(h -> 0) [f(x0 + h) − f(x0) − L(h)] / ||h|| = 0. In this case, L is the best linear approximation to f at x0 and is represented by the Jacobian matrix (or gradient in scalar-valued cases).

Difficulties with differentiability often arise at sharp corners or cusps; for example, the absolute value function is not differentiable at 0. Differentiability implies continuity, but the reverse is not true. Higher-order differentiability leads to smoothness concepts: C^k functions have continuous partial derivatives up to order k, C^∞ functions are infinitely differentiable, and analytic functions admit a local power series expansion.

Differentiability is central to optimization, geometry, and analysis, enabling tools such as the chain rule, Taylor expansions, and gradient-based methods. It is defined within the context of real and vector spaces equipped with norms or inner products.