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).

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.