This result is extremal with respect to the CFT since a consequence of the CFT is that, for any infinite recurrent word x, either the function is bounded, and in such a case x is periodic, or for infinitely many integers .
As a byproduct of the techniques used in the paper we extend a result of Harju and Nowotka (2002) in stating that any finite Fibonacci word , has only one critical point. Indeed we determine the exact number of critical points in any finite standard Sturmian word.