Differentierbarhet

Differentierbarhet, or differentiability, is a mathematical property of a function that expresses the existence of a linear approximation to the function at a point. Precisely, a function f from a subset of a normed space to another normed space is differentiable at a point a if there exists a bounded linear map L such that f(a+h) − f(a) − L(h) is small compared to the size of h as h tends to zero. The linear map L is called the differential or derivative of f at a, and in finite dimensions it is represented by the Jacobian matrix.

In one variable, differentiability at a point a means the limit (f(a+h) − f(a)) / h exists as h → 0, and the derivative f′(a) is that limit. Differentiability implies continuity, but the converse is not true: a function can be continuous at a point yet not differentiable there.

In several variables, differentiability means f(a+h) can be well approximated by f(a) plus a linear term in h. The matrix of partial derivatives at a, the Jacobian, provides this linear approximation. A common sufficient condition is that all partial derivatives exist in a neighborhood of a and are continuous at a; however, the existence of partial derivatives at a alone does not guarantee differentiability.

Common examples clarify the concept. The function f(x) = x^2 is differentiable everywhere; the absolute value f(x) = |x| is continuous but not differentiable at x = 0. In higher dimensions, functions like f(x,y) = sqrt(x^2 + y^2) fail to be differentiable at the origin.

Higher-order notions include C^k differentiability (k-times differentiable with continuous derivatives) and smoothness (C^∞). In some contexts, Fréchet differentiability and Gateaux differentiability describe related but distinct levels of differentiability in normed spaces.