At a Point of Differentiability, the Subdifferential of a Convex Function on an Open Convex Subset of $\mathbb{R}^n$ Is the Gradient Alone

In the setting of Convex Functions on $\mathbb{R}^n$: Standing Notation, let $U\subseteq\mathbb{R}^{n}$ be open and convex, let $f:U\to\mathbb{R}$ be convex, and let $x\in U$ be a point at which $f$ is differentiable, with gradient $\nabla f(x)$, defined by The Derivative Is the Jacobian Matrix, and Is Unique (i).

(i) The subdifferential is the gradient. The subdifferential of $f$ at $x$ is

$$\partial f(x)=\{\nabla f(x)\}.$$