A Convex Function on a Convex Subset of $\mathbb{R}^n$ Has a Subgradient at Every Interior Point

In the setting of Convex Functions on $\mathbb{R}^n$: Standing Notation, let $C\subseteq\mathbb{R}^{n}$ be convex, let $f:C\to\mathbb{R}$ be convex, and let $x\in\operatorname{int}C$ (so $x\in C$, by Closure and Interior in a Metric Space: Containments, the Closure is Closed, the Interior is Open (i)).

(i) Existence of a subgradient. The subdifferential $\partial f(x)$ is nonempty.