An example of nonuniqueness of a simple partial fraction of the best uniform approximation

For arbitrary natural $n\ge2$ we construct an example of a real continuous function, for which there exist more than one simple partial fraction of order $\le n$ of the best uniform approximation on a segment of the real axis. We prove that even the Chebyshev alternance consisting of $n+1$ points does not guarantee the uniqueness of the best approximation fraction. The obtained results are generalizations of known nonuniqueness examples constructed for $n=2,3$ in the case of simple partial fractions of an arbitrary order $n$.