nondifferentiable

Nondifferentiable describes a function or a point where the derivative does not exist. A function f is differentiable at a point a if the limit (f(a+h) - f(a))/h as h approaches 0 exists and is finite; the derivative at a is that limit. If the limit fails to exist, f is nondifferentiable at a. A function can be differentiable at some points and nondifferentiable at others, or nowhere differentiable on its domain.

Nondifferentiability can result from a cusp or corner (a sharp change in slope), a vertical tangent (the derivative becomes infinite), a discontinuity, or highly oscillatory behavior near the point. Differentiability implies continuity, but continuity alone does not guarantee differentiability.

Classic examples include f(x) = |x|, which is not differentiable at 0 due to different left and right slopes; f(x) = x^(1/3) has an infinite slope at 0; and the Weierstrass function is continuous everywhere but differentiable nowhere.

In several variables, differentiability means there exists a linear map that best approximates the change of f near a point. The existence of partial derivatives does not by itself guarantee differentiability.

Nondifferentiability is a central concept in analysis and has implications for smoothness, approximation, and the study of irregular or fractal behavior in functions.