Nonsmoothness

Nonsmoothness refers to the property of a function or surface having points where it is not differentiable. A function can be nonlinear and smooth, but it may still fail to be differentiable at certain points. In calculus and analysis, smoothness typically means a function is differentiable to a desired degree; nonsmoothness indicates the absence of a derivative at one or more points.

Common sources of nonsmoothness include piecewise definitions, absolute values, and max or min operations, which create corners or cusps. Examples: f(x) = |x| is nondifferentiable at x = 0; f(x) = max(x, y) is nondifferentiable along the region where the maximizing argument changes; f(x) = sqrt(|x|) is nondifferentiable at x = 0.

In nonsmooth analysis, researchers study generalized derivatives that extend the concept of a derivative to nondifferentiable points. The convex case uses subgradients; more generally, Clarke subdifferentials or proximal subdifferentials provide sets of directions that support a local linear model. These tools enable optimization and sensitivity analysis despite nondifferentiability.

Practically, nonsmooth optimization relies on subgradient methods, bundle methods, or proximal algorithms, and sometimes on smoothing techniques that approximate a nonsmooth function by a smooth one. Differentiability almost everywhere is guaranteed by Rademacher’s theorem for Lipschitz functions, but the derivative may fail on a set of points that matters for optimization.

Nonsmoothness appears across disciplines: in economics, engineering, signal processing, control, and machine learning, with examples such as L1 regularization and ReLU activations in neural networks.