piecewise (original) (raw)

Formally speaking, we have the following:

Remarks.

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  If D is an interval or a ray on ℝ, then this finite partition can usually be done so that each “piece” is an interval or a ray.
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  If function f satisfies property P, then f satisfies P piecewise.
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  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 ℝ is piecewise continuous is the same thing as saying that 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.