The word “piecewise” is used widely in mathematics, primarily in the analysisMathworldPlanetmath of functions of a single real variable. Piecewise is typically applied to a set of mathematical properties on a function. Loosely speaking, a function satisfies a particular property “piecewise” if that function can be broken down into pieces (to be made precise later) so that each piece satisfies that particular property. However, to avoid potential problems with infinityMathworldPlanetmath, the number of pieces is generally set to be finite (particularly in the case when the domain is bounded). Another potential problem is that the function having this “piecewise” property (let’s call it P) usually fails to have this property P at certain boundary points of the pieces. To get around this technicality, and thus allowing a much wider class of functions to being “piecewise P”, pieces are re-defined so as to exclude these problematic “boundary points”.

Formally speaking, we have the following:

That a function f with domain D having “piecewise” property P means that there is a finite partitionPlanetmathPlanetmath of D:

D=D1Dn, with DiDj= for ij

such that the restrictionPlanetmathPlanetmathPlanetmath of f to the interior of each Di: fi:=fint(Di) has property P.


  • If D is an intervalMathworldPlanetmathPlanetmath or a ray on , then this finite partition can usually be done so that each “piece” is an interval or a ray.

  • If function f satisfies property P, then f satisfies P piecewise.

  • Conversely, if f satisfies property P piecewise and f satisfies P at the boundary points of each “piece” of the domain D, then f satisfies P.

For example, if P means continuity of a function, then to say that a function f defined on is piecewise continuous is the same thing as saying that can be partitioned into intervals and rays so that f is continuousMathworldPlanetmathPlanetmath in each of the intervals and rays.

Other commonly used terms involving the concept of “piecewise” are piecewise differentiableMathworldPlanetmathPlanetmath, piecewise smooth, piecewise linear, and piecewise constant.

Anyone who can supply some graphs illustrating the concepts mentioned above will be greatly appreciated.

Title piecewise
Canonical name Piecewise
Date of creation 2013-03-22 15:50:42
Last modified on 2013-03-22 15:50:42
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Definition
Classification msc 26A99