Differenciálhatója

Differenciálhatója is a Hungarian morphological form derived from the adjective differenciálható, meaning differentiable. In mathematical writing the standard noun for the property is differenciálhatóság, while differenciálhatója is a possessive/inflected form that may appear in sentences referring to “its differentiable aspect” or in less formal usage. The term does not denote a separate mathematical concept beyond differentiability itself, but rather a grammatical way to reference the differentiable property of a specific object.

In analysis, a function f defined on a real or vector domain is differentiable at a point x0 if there exists a linear map L such that

f(x0 + h) − f(x0) − L(h) = o(∥h∥) as h → 0.

The linear map L is the derivative at x0, denoted f′(x0) in one dimension or the Jacobian in higher dimensions. If a function is differentiable on an open set, it is continuous there; differentiability is a stronger condition than mere continuity and does not follow from the existence of partial derivatives alone.

Differentiability can be extended to higher orders: a function is of class C^k if it has continuous derivatives up to order k, and smooth (C^∞) or analytic if derivatives of all orders exist with stronger convergence properties. Common examples illustrate the concept: f(x) = x^2 is differentiable for all x with derivative f′(x) = 2x, whereas f(x) = |x| is not differentiable at x = 0.

See also: differentiability, differentiable functions, higher-order differentiability, smoothness.