Piecewise Chebyshev approximation, Part 2: Approaching analytical solution

Let's go back to our problem. Let's assume that our  is monotonic function, that can have first-kind discontinuities. It's pretty simple to show that in this case for optimal solution all height of function graphic will be filled by intervals of height . So, we can knowing value of and values of function at the edges of interval we can compute number of parts in partition: , and knowing number of parts we can simply compute our optimal deviations: . We can say that for monotonic functions the problem is solved analytically. Moving further with analytic solution we'll take a look at the case when given function has only one extremum at and is continuous in some neighborhood of this extremum.