Differences and Constants for Functions of Class $C^2$ on a Euclidean Open Set

Let $n$ be a natural number, let $U\subseteq\mathbb{R}^n$ be an open subset of Euclidean space $\mathbb{R}^n$, and let $\mathbb{R}$ be the set of real numbers with the operations and the order $\le$ of its ordered field structure, where for $s,t\in\mathbb{R}$ we write $s<t$ to mean that $s\le t$ and $s\ne t$, and $s-t$ abbreviates $s+(-t)$.

Let $u,\varphi:U\to\mathbb{R}$ be of class $C^2$ on $U$, and let $u-\varphi:U\to\mathbb{R}$ be the function whose value at $y\in U$ is $u(y)-\varphi(y)$. For $c\in\mathbb{R}$ let $k_c:U\to\mathbb{R}$ be the function whose value at every $y\in U$ is $c$.

Then the following hold.

1. (Differences) The function $u-\varphi$ is of class $C^2$ on $U$, and for every $x\in U$ the gradients and Hessian matrices at $x$ satisfy

where the first difference is the difference of points of $\mathbb{R}^n$ and the second is the difference of real matrices.

2. (Constants) For every $c\in\mathbb{R}$ the function $k_c$ is of class $C^2$ on $U$, and for every $x\in U$ the gradient $Dk_c(x)$ is the origin of $\mathbb{R}^n$, that is, the point all of whose coordinates are $0$, and the Hessian matrix $D^2k_c(x)$ is the real $n\times n$ matrix all of whose entries are $0$.