Restriction of a linear map on a compact convex subset of $\mathbb R^n$ with non-empty interior

Question

Let $f: \Omega \to \mathbb{R}$, where $\Omega$ is a convex compact subset of $\mathbb{R}^n$ with a non-empty interior, denoted by $Int(\Omega)$. Assume that on $Int(\Omega)$, $f$ is the restriction of a linear function $L: \mathbb{R}^n \to \mathbb{R}$. Is $f$ necessarily the restriction of $L$ to $\Omega$ ?

Note that $f$ is not assumed continuous on the frontier of $\Omega$.

Thank you very much for your help.