Slice Function and the Partial Derivative

Let $n$ be a natural number, let $U\subseteq\mathbb{R}^n$ be an open subset of Euclidean space $\mathbb{R}^n$, let $f:U\to\mathbb{R}$ with $\mathbb{R}$ the set of real numbers, let $a=(a_1,\dots,a_n)\in U$, and let $i\in\{1,\dots,n\}$. Let $|\cdot|$ be the absolute value on $\mathbb{R}$, and for $s\in\mathbb{R}$ let $a[s]$ denote the point of $\mathbb{R}^n$ whose $i$th coordinate is $s$ and whose $k$th coordinate is $a_k$ for every $k\in\{1,\dots,n\}$ with $k\ne i$, so that $a[a_i]=a$. Then the following hold.

1. (Admissible radius) There exists $\rho\in\mathbb{R}$ with $0<\rho$ such that $a[s]\in U$ for every $s\in\mathbb{R}$ with $|s-a_i|<\rho$.

2. (Slice function) Let $\rho$ be as in claim 1, let

which is an interval having $a_i$ as an interior point, and let $g:I\to\mathbb{R}$ be the slice function of $f$ at $a$ in the $i$th variable, given by $g(s)=f(a[s])$.

Then the partial derivative of $f$ with respect to the $i$th variable at $a$ exists if and only if $g$ is differentiable at $a_i$, and in that case