differentiables

Differentiability is a local smoothness property of a function. A function f defined on an open subset of R^n is differentiable at a point a if there exists a linear map L such that f(a+h) = f(a) + L(h) + o(||h||) as h → 0. The linear map L is the derivative Df(a); in one dimension it is the usual f'(a), and in coordinates it is the Jacobian matrix. Intuitively, differentiability means f admits a best linear approximation near a.

In one variable, f is differentiable at a if the limit lim h→0 [f(a+h) − f(a)]/h exists. Differentiability implies continuity at a, but the converse is false (e.g., f(x) = |x| is continuous yet not differentiable at 0).

In multiple variables, differentiability means the map can be locally approximated by its linear differential df_a. If f: R^n → R, the gradient ∇f(a) is the vector representation of Df(a).

Rules: If f is differentiable at a and g is differentiable at f(a), then g∘f is differentiable at a, with the chain rule D(g∘f)(a) = Dg(f(a)) ∘ Df(a). The product and quotient rules hold where defined.

Higher regularity is described by C^k and C^∞ classes: C^1 means the first derivatives exist and are continuous; C^∞ means derivatives of all orders exist and are continuous. Differentiability does not guarantee continuity of the derivative.

Remarks: In complex analysis, differentiability (holomorphy) is a stronger condition than in real analysis. Some results, such as Rademacher’s theorem, state that Lipschitz functions are differentiable almost everywhere.