  Let $L:[0,1]\setminus\{d\}\rightarrow [0,1]$ be a one-dimensional Lorenz like
expanding map ($d$ is the point of discontinuity), $\mathcal{P}=\{ (0,d),(d,1)
\}$ be a partition of $[0,1]$ and $C^{\alpha}([0,1],\mathcal{P})$ the set of
piecewise H\"older-continuous potential of [0,1] with the usual $\mathcal{C}^0$
topology. In this context, we prove, improving a result of \cite{BS03}, that
piecewise H\"older-continuous potential $\phi$ satisfying \linebreak
$\max\left\{ \limsup_{n \rightarrow \infty}\frac{1}{n}(S_{n}\phi)(0),\limsup_{n
\rightarrow \infty}\frac{1}{n}(S_{n}\phi)(1)\right\}<P_{\text{top}}(\phi,T)$
support an unique equilibrium state. Indeed, we prove there exists an open and
dense subset $\mathcal{H}$ of $C^{\alpha}([0,1],\mathcal{P})$ such that, if
$\phi \in \mathcal{H}$, then $\phi$ admits one equilibrium measure.
