Piecewise

In mathematics, a function f(x) of a real number variable x is defined piecewise, if f(x) is given by different expressions on various intervals. An example is given by the Heaviside step function.

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According to the standard definitions, this is a single function, that happens to have its value computed by different methods in different cases. It is useful to do this, for example to make a sawtooth function. That is an example of a piecewise linear function: its graph is made up of a number of parts of the graphs of linear functions. Problems can arise at the ends of the intervals used for separate definitions. We must give a definite value for f(x) there, as everywhere else. It may be a point where continuity fails (as for the Heaviside function at 0), or where the function isn't smooth (the absolute value function at 0).

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The definitions of piecewise continuous, piecewise differentiable and so on are therefore made, to require that the 'pieces' of the function are continuous (resp. differentiable), but that at the end points failure of those conditions is allowed. A path said to be piecewise continuously differentiable is a continuous path (in the plane, say) but which can at some points turn direction sharply, so the continuity of the derivative vector at those points doesn't hold.

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