Piecewiseconstant

Piecewiseconstant refers to a class of functions that are constant on each member of a partition of their domain. More precisely, let I be an interval and {I_k} a partition of I into pairwise disjoint subintervals such that I = ⋃ I_k. A function f is piecewiseconstant if there exist constants c_k with f(x) = c_k for all x in I_k. The partition can be finite or countably infinite, and when endpoints are shared between subintervals, one must specify whether f is left- or right-continuous at those points; the Heaviside step function is a canonical single-jump example.

A convenient representation is f(x) = ∑_k c_k · χ_{I_k}(x), where χ_{I_k} is the indicator function of I_k. This emphasizes that piecewiseconstant functions are simple in structure: constant on each region of the partition, with possible jumps at the region boundaries.

Key properties include that all discontinuities are jumps at the partition boundaries, and such functions are locally integrable and have bounded variation when the number of jumps is finite. They can be extended to more general domains by using measurable sets instead of intervals, yielding simple functions used in measure theory.

Applications of piecewiseconstant functions are common in numerical analysis and signal processing. They arise as stair-step approximations to more complex functions, in discretized models, and in digital representations where a quantity takes constant values over ranges of the input. They also serve as foundational examples in the study of piecewise-defined and indicator-based constructions.