deriverbare

Deriverbare is a term used in calculus to describe functions that have a derivative at points in their domain. In the single-variable case, a function f: D → R is deriverbar on D if the derivative f'(x) exists for every x in D. The derivative at x is defined by the limit f'(x) = lim_{h→0} (f(x+h) − f(x)) / h, when this limit exists. In several variables, a function f: D ⊆ R^n → R^m is deriverbar at a point a if there is a linear map L: R^n → R^m such that lim_{h→0} (f(a+h) − f(a) − L(h)) / ||h|| = 0; L is the derivative, also called the Jacobian or tangent map.

If a function is deriverbar on an open set, it is continuous on that set and possesses a linear approximation at every point provided by its derivative. Differentiability implies continuity, but continuity does not imply differentiability. Common examples illustrate the distinction: f(x) = x^2 is deriverbar everywhere with f'(x) = 2x; f(x) = |x| is not deriverbar at x = 0, since the left and right derivatives do not agree.

In higher dimensions, the derivative at a point is represented by a Jacobian matrix or a linear map that best approximates the function near that point. The concept underpins results such as the chain rule, product rule, and the mean value theorem (the latter requiring differentiability on an interval). Differentiability is stronger than mere existence of a derivative along a coordinate direction and is central to the study of smoothness classes, such as C^1, which require continuous derivatives.