|
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 function.
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.
See also
|