$C^2$ Real-Valued Map on an Open Subset of Euclidean Space

Let $n$ be a natural number, let $U\subseteq\mathbb{R}^n$ be an open subset of Euclidean space $\mathbb{R}^n$, and let $f:U\to\mathbb{R}$, where $\mathbb{R}$ is the set of real numbers.

We say that $f$ is of class $C^2$ on $U$ if the following two conditions hold.

1. $f$ is of class $C^1$ on $U$, regarded there as a map into $\mathbb{R}^m$ with $m=1$ and single coordinate function $f$.

2. For every $i\in\{1,\dots,n\}$, the function from $U$ to $\mathbb{R}$ whose value at $x\in U$ is the partial derivative of $f$ with respect to the $i$th variable at $x$, denoted $\partial f/\partial x_i$, is of class $C^1$ on $U$.

If $f$ is of class $C^2$ on $U$, then for all $i,j\in\{1,\dots,n\}$ we write

for the function from $U$ to $\mathbb{R}$ whose value at $x\in U$ is the partial derivative with respect to the $i$th variable, at $x$, of the function $\partial f/\partial x_j$.