Piecewise function

In mathematics, a piecewise function (also called a piecewise-defined function, a hybrid function, or a function defined by cases) is a function whose domain is partitioned into several intervals ("subdomains") on which the function may be defined differently. Piecewise definition is actually a way of specifying the function, rather than a characteristic of the resulting function itself, as every function whose domain contains at least two points can be rewritten as a piecewise function. The first three paragraphs of this article only deal with this first meaning of "piecewise".

Terms like piecewise linear, piecewise smooth, piecewise continuous, and others are also very common. The meaning of a function being piecewise

P

{\displaystyle P}

, for a property

P

{\displaystyle P}

, is roughly that the domain of the function can be partitioned into pieces on which the property

P

{\displaystyle P}

holds, but this term is used slightly differently by different authors. Unlike the first meaning, this is a property of the function itself and not only a way to specify it. Sometimes the term is used in a more global sense involving triangulations; see Piecewise linear manifold.

Notation and interpretation

Piecewise functions can be defined using the common functional notation, where the body of the function is an array of functions and associated subdomains. A semicolon or comma may follow the subfunction or subdomain columns. The

if

{\displaystyle {\text{if}}}

or

for

{\displaystyle {\text{for}}}

is rarely omitted at the start of the right column.