Piecewisesmooth

Piecewisesmooth, often written as piecewise-smooth, denotes a class of functions that are smooth on each piece of a finite partition of their domain but may fail to be smooth at the partition points. A common formalization is: a function f: [a,b] → R is piecewisesmooth if there exists a finite sequence a = x0 < x1 < ... < x_m = b such that on each open interval (x_{i-1}, x_i) the function is C^k for some k (typically k = 1 or k = 2), and f is continuous on [a,b]. At the breakpoints the function may not be differentiable, and derivatives can have jump discontinuities, though the exact continuity requirements at the breakpoints vary by context.

In practice, piecewisesmooth is used to describe functions that are smooth on each subinterval of a partition, with finite many non-smooth points. The most common conventions assume piecewisesmooth functions are continuous on the domain and have a finite number of nondifferentiable points; some definitions allow a jump in the function value at breakpoints, but many applications emphasize continuity.

Examples help illustrate the idea. The absolute value function |x| on [-1,1] is piecewise smooth: it is smooth on (-1,0) and (0,1) with a cusp at 0 and a jump in the derivative, while remaining continuous. Another example is f(x) = { x^2 for x ≤ 0, x^3 for x > 0 }, which is continuous and smooth on each side of 0 but not differentiable in a unique, shared tangent at the origin.

Applications of piecewisesmooth functions appear across numerical analysis, computer graphics, and applied mathematics. They model curves and surfaces that are smooth within regions but have corners or kinks at known boundaries, and they arise in piecewise-defined dynamical systems and in spline-based representations. Related concepts include piecewise linear functions, splines, and non-smooth analysis.