nondifferentiability

Nondifferentiability is the absence of a derivative at a point for a real-valued function. A function f defined on an interval is differentiable at a if the limit lim_{h→0} (f(a+h) − f(a))/h exists and is finite; this limit, when it exists, is the derivative f′(a). If f is differentiable at a, it is necessarily continuous at a.

Nondifferentiability can arise from several phenomena. A cusp or sharp corner occurs when left and right slopes do not agree. A vertical tangent happens when the slope becomes infinite as the point is approached. Rapid oscillation near the point can prevent a well-defined tangent. Discontinuities also yield nondifferentiability. At endpoints of an interval, differentiability is defined using one-sided derivatives.

Examples illustrate these ideas. The function f(x) = |x| is not differentiable at 0 because the left and right derivatives are −1 and +1. The function f(x) = x^{1/3} is not differentiable at 0 since the slope tends to infinity from both sides. The function f(x) = sin(1/x) for x ≠ 0 with f(0) = 0 is not differentiable at 0 due to an oscillatory difference quotient. The function f(x) = sqrt(|x|) has a vertical tangent at 0. There are continuous functions that are nowhere differentiable, such as the Weierstrass function.

Related results and concepts include that Lipschitz functions are differentiable almost everywhere (Rademacher’s theorem) and that nondifferentiability motivates tools from nonsmooth analysis, such as subdifferentials, which generalize the derivative to certain non-smooth contexts.

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