\bigl\{a \ge a_0 \bigr\} \\
& U_{\p}(t_0,a_0)=\bigl\{(t,a)\in \cal P~|~-\max \p \le \frac{t-t_0}{a-a_0} \le -\min \p \bigr\} \cap \bigl\{a \ge a_0 \bigr\}\\
& L_{\p}^-(t_0,a_0)=\bigl\{(t,a)\in \cal P~|~ \frac{t-t_0}{a-a_0} > -\min \p \bigr\} \cap
\bigl\{a < a_0 \bigr\} \\
& L_{\p}^+(t_0,a_0)=\bigl\{(t,a)\in \cal P~|~\frac{t-t_0}{a-a_0} < -\max \p \bigr\} \cap
\bigl\{a < a_0 \bigr\} \\
& L_{\p}(t_0,a_0)=\bigl\{(t,a)\in \cal P~|~ -\max \p \le \frac{t-t_0}{a-a_0} \le  -\min \p \bigr\} \cap \bigl\{a < a_0 \bigr\}\end{aligned}$$ Then we define the region $S_{\p}(t_0,a_0)=U_{\p}(t_0,a_0) \cup L_{\p}(t_0,a_0)$, the sectors $S_{\p}^-(t_0,a_0)=U_{\p}^-(t_0,a_0) \cup L_{\p}^-(t_0,a_0)$ and $S_{\p}^+(t_0,a_0)=U_{\p}^+(t_0,a_0) \cup L_{\p}^+(t_0,a_0)$.

\[lemma\_sector\] Suppose that $(t,a)\in \cal P$ . Then

-   $\trans(F_{t,a})\le \trans(F_{t_0,a_0})$ if $(t,a) \in S_{\p}^-(t_0,a_0)$ and

-   $\trans(F_{t,a})\ge \trans(F_{t_0,a_0})$ if $(t,a) \in S_{\p}^+(t_0,a_0)$.

The inequalities are strict if $\trans(F_{t_0,a_0})$ is irrational.

Note that $F_{t,a}(x)-F_{t_0,a_0}(x)=(t-t_0)+(a-a_0)\p(x)$. The way the sectors $S_{\p}^-(t_0,a_0)$ and $S_{\p}^+(t_0,a_0)$ are defined, we see that if $(t,a) \in S_{\p}^-(t_0,a_0)$ then $F_{t,a}< F_{t_0,a_0}$ and if $(t,a)\in S_{\p}^+(t_0,a_0)$ then $F_{t,a} > F_{t_0,a_0}$. Hence the result follows by Lemma \[lemma\_transincrease\].

Let us prove the Lipschitz continuity of $\gamma_{p/q}^l$ first. Assume that $t_0=\gamma_{p/q}^l(a_0)$.

First, according to Lemma \[lemma\_sector\], if $(t_1,a_1) \in S_{\p}^+(t_0,a_0)$ then $\trans(F_{t_1,a_1}) \ge p/q= \trans(F_{t_0,a_0})$.

Second, if $(t_1,a_1) \in S_{\p}^-(t_0,a_0)$ then there is a point $(t_2,a_0)$ in $S_{\p}^+(t_1,a_1)$ with $t_2<t_0$. According to Lemma \[lemma\_sector\], $\trans(F_{t_1,a_1})\leq \trans(F_{t_2,a_0})$. Since 