$t_2<t_0$ and $(t_0,a_0)$ is on the left boundary of the Arnol’d tongue $\cal T_{p/q}$, $\trans(F_{t_2,a_0})<\trans(F_{t_0,a_0})=p/q$. Consequently $\trans(F_{t_1,a_1})<p/q$.

It follows that if $t_1=\gamma_{p/q}^l(a_1)$, then $(t_1,a_1)\in S_{\p}(t_0,a_0)$. This shows the Lipschitz continuity of $\gamma_{p/q}^l$: $$-\max \p \le \frac{\gamma_{p/q}^l(a_1)-\gamma_{p/q}^l(a_0)}{a_1-a_0} \le -\min \p.$$ Similarly one can prove that $$-\max \p \le \frac{\gamma_{p/q}^r(a_1)-\gamma_{p/q}^r(a_0)}{a_1-a_0} \le -\min \p.$$ These complete the proof of the Lipschitz continuities of $\gamma_{p/q}^{l}$ or $\gamma_{p/q}^r$.

Since $\gamma_{p/q}^l(0)=p/q=\gamma_{p/q}^r(0)$ we have the following corollary.

When $a \rightarrow 0$, we have $\gamma_{p/q}^{l}(a) \rightarrow p/q$ and $\gamma_{p/q}^r(a)\to p/q$.

If for a nonzero $a_0$, $a=a_0$ intersects $\cal T_{p/q}$ on a closed interval of positive length then $\cal T_{p/q}$ has non empty interior and thus it is of positive area.

By construction $\cal T_{p/q}$ is bounded by $\gamma_{p/q}^l$ and $\gamma_{p/q}^r$. Also by assumption $a=a_0$ intersects $\cal T_{p/q}$ on a closed interval say $I_{a_0}(p/q)$, of positive length. Then by the continuity of $\gamma_{p/q}^l$ and $\gamma_{p/q}^r$ there is a non empty open neighbourhood of the interior $I_{a_0}(p/q)^{\circ}$ bounded inside $\cal T_{p/q}$. Which proves that $\cal T_{p/q}$ has non empty interior and it is of positive area.

Now we show that the irrational tongues are also Lipschitz continuous.

For an irrational $\alpha$, the function $\gamma_{\alpha}:(a_{\min},a_{\max})\to
\R$ is Lipschitz continuous. More precisely, 