Machinery & Energetics

In this paper we consider the problem for continuous functions with a finite number of extremal points defined on an interval and a numerical axis. It is shown that in each equivalence class of such functions there is a nonnegative function that takes at the extremum points all integer values from the set 0, 1, 2, ..., l. For a complete alternating sequence, a piecewise linear function is defined, which is called a PL-realization of an alternating sequence. It is proved that every continuous function with a finite number of extremal points, given on an interval, will be topologically equivalent to PL-realization of its complete alternating sequence. A periodic alternating sequence is defined which is constructed according to the sequence of extrema of a continuous function defined on the interval [a, b] and numbers corresponding to the critical values of the given function. A special function is introduced and a special alternating function is assigned to the special function. The existence of a polynomial topologically equivalent to a piecewise-linear function is proved.

alternating sequence, extremum, topological equivalence, polynomial, piecewise-linear function