Vuillon from [@Vu] is important: a word $\uu$ is Sturmian if and only if any prefix of $\uu$ has exactly two return words. By combining this result with [@Durand98], we obtain an essential observation about derivated words of Sturmian words, which also follows from [@AraBru05].

\[thm:derstst\] If $\uu$ is a Sturmian word and $w$ is a prefix of $\uu$, then its derivated word $d_{\uu}(w)$ is Sturmian as well.

Set ${\bf v} = d_{\bf u}(w)$. Let $p$ be a prefix of ${\bf v}$. Due to Proposition 2.6 in [@Durand98], there exists a prefix $q$ of $\uu$ such that $d_{\bf v}(p) = d_{\bf u}(q) $. By Vuillon’s characterization of Sturmian words, the word $ d_{\bf u}(q)$ is binary. It means that any prefix $p$ of ${\bf v} $ has two return words in ${\bf v} $ and so ${\bf v} $ is Sturmian.

The Sturmaian words (sequences) were originally defined by Hedlund and Morse in [@HeMo]. Their definition is more general as they consider also biinfinite words and (in terms of our definition above) rational slopes. Hence their Sturmian words may not be recurrent. For details on the history of definition of Sturmian words see [@Fogg], especially the historical remark at page 