# piecewise

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 $P$) 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 $D\subseteq\mathbb{R}$ having “piecewise” property $P$ means that there is a finite partition  of $D$:

 $D=D_{1}\cup\cdots\cup D_{n},\mbox{ with }D_{i}\cap D_{j}=\varnothing\mbox{ for% }i\neq j$

such that the restriction   of $f$ to the interior of each $D_{i}$: $f_{i}:=f\mid\operatorname{int}(D_{i})$ has property $P$.

Remarks.

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

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

Title piecewise Piecewise 2013-03-22 15:50:42 2013-03-22 15:50:42 CWoo (3771) CWoo (3771) 10 CWoo (3771) Definition msc 26A99