Piecewise

Piecewise describes a function or relation defined by multiple expressions, each applicable on a specific region of the input’s domain. A piecewise function is given by a set of rules, with the domain partitioned so that every input satisfies exactly one condition. For example, f(x) = { x^2 if x < 0; 2x if 0 ≤ x < 3; 5 if x ≥ 3 }.

Notation is typically shown with braces or a case-style listing and explicit conditions for each piece. At boundary points, the convention chosen in the definition determines the value there. It is common to assign a boundary to one piece (or, in some contexts, to make pieces meet so the function is continuous). A valid piecewise function must be defined for every x in its domain and must avoid overlapping conditions.

Graphically, each piece is drawn on its respective subinterval, resulting in a graph that may be continuous or have corners, jumps, or other discontinuities at the transition points. The derivative, if it exists, can be discontinuous at those points even when the function itself is continuous. Piecewise functions cover a wide range of shapes, from piecewise linear to piecewise polynomial or constant functions, and can model abrupt changes in behavior.

Common examples include the absolute value function, defined as |x| = x if x ≥ 0 and −x if x < 0, and various “if-then” models used in economics, physics, and computer science. In mathematics, defining by cases or a case distinction is the standard terminology for constructing such functions.