for all $a_0$ and $a_1$ in $(a_{\min}, a_{\max})$, $$-\max \p \le \frac{\gamma_{\alpha}(a_1)-\gamma_{\alpha}(a_0)}{a_1-a_0} \le -\min \p.$$

In this case we shall concentrate on the irrational tongue $\cal T_{\alpha}$ and the function $\gamma_{\alpha}$. The proof of the fact that $\gamma_{\alpha}$ is Lipschitz is similar to that of Proposition \[prop\_lipschbd\]. For a fixed $a_0\in (a_{\min},a_{\max})$ let $(t_0,a_0)$ be a point on $\gamma_{\alpha}$. From Definition \[def\_sector\] we have the sectors $S_{\p}^-(t_0,a_0), S_{\p}^+(t_0,a_0)$ and the region $S_{\p}(t_0,a_0)$ defined. According to Lemma \[lemma\_sector\] we see that $\trans(F_{t,a})<\trans(F_{t_0,a_0})=\alpha$ if $(t,a)\in S_{\p}^-(t_0,a_0)$ and $\trans(F_{t,a})>\trans(F_{t_0,a_0})=\alpha$ if $(t,a)\in S_{\p}^+(t_0,a_0)$. Thus $\trans$ takes the value $\alpha$ in the region $S_{\p}(t_0,a_0)$. Hence $$-\max \p \le \frac{\gamma_{\alpha}(a)-\gamma_{\alpha}(a_0)}{a-a_0} \le -\min \p.$$

The Lipschitz continuity of $\gamma_{p/q}^l$ and $\gamma_{p/q}^r$ confirms that $\gamma_{p/q}^l$ and $\gamma_{p/q}^r$ define continuous curves in the parameter space $\cal P$. The same holds true for the curve $\gamma_{\alpha}$. In fact we can prove more when we are in the analytic standard family.

\[theo\_bdrystdanalytic1st\] In the standard family $(S_{t,a})_{(t,a)\in \cal P_S}$, the boundary curves of $\cal T_{p/q}$ are analytic functions of $a$ for $a\in (-1/2\pi,1/2\pi)\setminus \{0\}$.

Suppose that $a_0\ne 0$ is a point in $(-1/2\pi,1/2\pi)\setminus \{0\}$. We shall show that $\gamma_{p/q}^l$ and $\gamma_{p/q}^r$ are analytic around $a_0$. Let’s consider $\gamma_{p/q}^l$ first. There exist 