The proof we present here is simpler and we only assume that $\phi$ is Lipschitz. We also show that the irrational tongues are Lipschitz curves. Let us recall the definition of the of the tongues.

\[def\_bdry\] In the parameter space $\cal P$, the line $a=a_0$ intersects the rational tongue $\cal T_{p/q}$ on an interval $I_{a_0}(p/q)=[\gamma_{p/q}^l(a_0),\gamma_{p/q}^r(a_0)]$ and it intersects the irrational tongue $\cal T_{\alpha}$ for $\alpha\in \R\setminus\Q$ on a singleton set $\{\gamma_{\alpha}(a_0)\}$. Thus we have three type of functions, namely $\gamma_{p/q}^{l}:(a_{\min},a_{\max})\to \R$, $\gamma_{p/q}^r:(a_{\min},a_{max})\to \R$ and $\gamma_{\alpha}:(a_{\min},a_{\max})\to \R$. These functions define the of the tongues.

\[prop\_lipschbd\] The functions $\gamma_{p/q}^l$ and $\gamma_{p/q}^r: (a_{\min},a_{\max}) \to \R$ are Lipschitz continuous. More precisely for all $a_0$ and $a_1$ in $(a_{\min},a_{\max})$, $$\begin{aligned}
&-\max \p \le \frac{\gamma_{p/q}^l(a_1)-\gamma_{p/q}^l(a_0)}{a_1-a_0} \le -\min \p \quad \text{and}\\
&-\max \p \le \frac{\gamma_{p/q}^r(a_1)-\gamma_{p/q}^r(a_0)}{a_1-a_0} \le -\min \p.  \end{aligned}$$

![Definitions of the region $S_{\p}$ and the sectors $S_{\p}^-,S_{\p}^+$.](lipschitz_curve.pdf){width="4in" height="5in"}

To prove this we need the following definition and a lemma.

\[def\_sector\] Let us define some regions of the parameter space $\cal P$ around a base point $(t_0,a_0)$ in the following manner. $$\begin{aligned}
& U_{\p}^-(t_0,a_0)=\bigl\{(t,a)\in \cal P~|~ \frac{t-t_0}{a-a_0} < -\max \p \bigr\} \cap
\bigl\{a \ge a_0 \bigr\} \\
& U_{\p}^+(t_0,a_0)=\bigl\{(t,a)\in \cal P~|~ \frac{t-t_0}{a-a_0} > -\min \p \bigr\} \cap
