The word “piecewise” is used widely in mathematics, primarily in the analysis 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 infinity, 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 ) usually fails to have this property at certain boundary points of the pieces. To get around this technicality, and thus allowing a much wider class of functions to being “piecewise ”, pieces are re-defined so as to exclude these problematic “boundary points”.

Formally speaking, we have the following:

That a function with domain having “piecewise” property means that there is a finite partition of :

with for

such that the restriction of to the interior of each : has property .

Remarks.

• If is an interval or a ray on , then this finite partition can usually be done so that each “piece” is an interval or a ray.
• If function satisfies property , then satisfies piecewise.
• Conversely, if satisfies property piecewise and satisfies at the boundary points of each “piece” of the domain , then satisfies .

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

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

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