Piecewise-Defined Function

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A Piecewise-Defined Function is a function composed of subfunctions that apply to non-overlapping intervals of the function's domain.



References

2012

  • http://en.wikipedia.org/wiki/Piecewise
    • In mathematics, a piecewise-defined function (also called a piecewise function) is a function which is defined by multiple subfunctions, each subfunction applying to a certain interval of the main function's domain (a subdomain). Piecewise is actually a way of expressing the function, rather than a characteristic of the function itself, but with additional qualification, it can describe the nature of the function. For example, a piecewise polynomial function: a function that is a polynomial on each of its subdomains, but possibly a different one on each.

      The word piecewise is also used to describe any property of a piecewise-defined function that holds for each piece but may not hold for the whole domain of the function. A function is piecewise differentiable or piecewise continuously differentiable if each piece is differentiable throughout its subdomain, even though the whole function may not be differentiable at the points between the pieces. In convex analysis, the notion of a derivative may be replaced by that of the subderivative for piecewise functions. Although the "pieces" in a piecewise definition need not be intervals, a function isn't called "piecewise linear" or "piecewise continuous" or "piecewise differentiable" unless the pieces are intervals.