smoothness

Smoothness is a qualitative and quantitative notion describing how free a object or function is from roughness or abrupt changes. In mathematics, smoothness nearly always refers to differentiability properties of a function or a geometric object. A function is called smooth if it has derivatives of all orders that are continuous; such a function is often denoted as being of class C^∞. If only finitely many derivatives exist or are continuous, the function is said to be of class C^k. We also use weaker, yet related, notions of smoothness such as Lipschitz continuity (bounded rate of change), and Hölder or C^{k,α} continuity, which allows controlled but nonzero fractional smoothness. In many areas, fractional smoothness is captured by Sobolev spaces H^s or Besov and Triebel–Lizorkin spaces, which quantify smoothness in an integral or frequency sense.

In geometry, a smooth manifold is a space that, locally, looks like Euclidean space and has smooth transition maps between charts. A smooth surface has well-defined tangent planes and curvature almost everywhere, with no sharp creases or edges.

Examples help illustrate the idea: polynomials and trigonometric functions are smooth; the absolute value function or the distance to the origin in Euclidean space fail to be differentiable at certain points, hence are not smooth there.

Applications of smoothness appear across mathematics and applied sciences. In numerical analysis, higher smoothness often yields faster convergence of approximation schemes; in signal processing and graphics, smooth signals and surfaces produce better interpolation and rendering; in statistics and data analysis, smoothing kernels and regularization frameworks rely on smoothness assumptions about the underlying truths.