Nondifferentiation

Nondifferentiation, or nondifferentiability, is the property of a function not having a derivative at a given point. A function f is differentiable at x0 if the limit lim_{h→0} (f(x0+h)−f(x0))/h exists and is finite; if this limit fails to exist, f is nondifferentiable at x0. Differentiability is a local property and implies continuity, but a continuous function need not be differentiable.

Common causes include sharp corners, cusps, vertical tangents, and discontinuities. For example, f(x)=|x| is differentiable for all x≠0 and has a cusp at x=0, so it is nondifferentiable there. The function f(x)=x^{1/3} has a vertical tangent at x=0 and is nondifferentiable there. There are functions that are continuous everywhere but differentiable nowhere; the classical Weierstrass function is an example of this behavior.

In higher dimensions, nondifferentiability means the differential does not exist at a point; differentiability implies the existence of a linear approximation by the gradient. A related result is Rademacher's theorem: every Lipschitz function on Euclidean space is differentiable almost everywhere. Nondifferentiability thus highlights the boundary between smooth and irregular behavior in real and multivariable analysis.