map $F_{t,a}$ induces an orientation preserving circle homeomorphism $f_{t,a}$. A well known example [@Arnold] is the or the of the form $$S_{t,a} : x\mapsto x+t+a \sin (2\pi x).$$ The parameter space of the standard family is $\cal P_S=\{(t,a)~|~t\in \R, -1/2\pi <a < 1/2\pi \}.$ Arnol’d studied this using the translation number and the rotation number of the map corresponding to each point in the parameter space. For $\theta \in \R$ we define the $$\mathcal T_{\theta} \eqdef \{(t,a)\in \cal P~|~\trans(F_{t,a})=\theta\}.$$ The collection $\{\mathcal T_\theta\}_{\theta \in \R}$ gives a partition of the parameter space $\cal P$. In this article we shall study the boundaries of these Arnol’d tongues and this is inspired after the work of Arnol’d, Herman, Hall, Boyland and others (see [@Arnold], [@Herbig], [@Hall], [@boyland]).\
For a rational translation number $p/q$, when $p$ and $q$ are coprime and for a fixed $a\in (a_{\min},a_{\max})$, the set of values of $t$ such that $(t,a) \in \mathcal T_{p/q}$ is an interval $\left[\gamma_{p/q}^l(a),\gamma_{p/q}^r(a)\right]$. And for an irrational $\alpha$ and a fixed $a \in (a_{\min},a_{\max})$ the set of $t$ such that $(t,a) \in \cal T_{\alpha}$ is singleton $\{\gamma_{\alpha}(a)\}$. These define the and three functions $\gamma_{p/q}^{l} : (a_{\min},a_{\max}) \to \R$, $\gamma_{p/q}^{r} : (a_{\min},a_{\max}) \to \R$ and 