A Convex Function is Continuous near an Interior Point

Let $n$, $[n]$, $\mathbb{R}$, $\mathbb{R}^{n}$, the Euclidean norm $\lVert\cdot\rVert$, the Euclidean metric $d_{E}$ and the closed balls $\bar{B}_{d_{E}}(x,s)$ of Closed Ball in a Metric Space be as in A Convex Function is Lipschitz on a Ball around an Interior Point. Regard $\mathbb{R}$ as a metric space with the absolute-value metric $d_{\mathbb{R}}$ of The Absolute Value Metric on the Real Line, so that $d_{\mathbb{R}}(a,b)=|a-b|$.

Let $C\subseteq\mathbb{R}^{n}$ be convex, let $u:C\to\mathbb{R}$ be convex on $C$, and let $x_{0}$ be an interior point of $C$ in $\mathbb{R}^{n}$, the topology on $\mathbb{R}^{n}$ being the one used in A Convex Function is Lipschitz on a Ball around an Interior Point.

Then there is $\rho\in\mathbb{R}$ with $0<\rho$ such that $\bar{B}_{d_{E}}(x_{0},\rho)\subseteq C$ and the restriction of $u$ to $\bar{B}_{d_{E}}(x_{0},\rho)$ is continuous on $\bar{B}_{d_{E}}(x_{0},\rho)$.