Let $X$ be a smooth projective curve, $\mu$ a positive Borel meausre on $X^{\textup{an}}$ and $\varphi$ a solution to the Monge-Ampère equation $c_{1}(L,\|\cdot\|e^{-\varphi})=\mu$. Let $\tau$ be an open face of a skeleton associated to a strictly semistable formal model of $X^{\textup{an}}$ on which $(L,\|\cdot\|)$ has a formal model. Suppose that $\mu$ is supported on that skeleton and is given on $\tau$ by $f\cdot\boldsymbol{dx}$ where $\boldsymbol{dx}$ denotes the Lebesgue measure on $\tau$. If $f\in C^{k}(\tau)$ then we have $\varphi\Big|_{\tau}\in C^{k+2}(\tau)$.