piecewisedefined

PiecewiseDefined, or piecewise-defined, describes a function that is defined by different expressions on different parts of its domain. Such functions are common in mathematics and are used to model situations where a single rule does not apply uniformly across all inputs. The term may appear in textbooks, lecture notes, or online resources, and in some cases the concatenated form piecewisedefined is seen as a keyword or handle in search queries and domain names.

Notation and presentation often use case notation or braces to list the rules. A typical form is: f(x) = { x^2 if x < 0; 2x + 1 if x ≥ 0 }. Each line pairs a rule with a condition on x. This structure makes it straightforward to describe functions with different behavior on intervals, at a boundary point, or on separate domains. Graphically, the function may have a break, a cusp, or a discontinuity where the defining rule changes.

Key considerations for piecewise-defined functions include continuity and differentiability at the transition points. The function is continuous at a boundary if the left- and right-hand limits agree and equal the function value there. Differentiability depends on whether the left and right derivatives match. In many real-world models, piecewise-defined functions represent threshold effects, saturation, or regime changes, such as tax brackets, piecewise-linear approximations, or signal processing steps.

See also: piecewise function, Heaviside step function, discontinuity, piecewise-linear modeling.