Derivatives Along a Segment for $C^1$ Functions on a Euclidean Open Set

Let $n$ be a natural number, let $W$ be an open subset of Euclidean space $\mathbb{R}^n$, let $\mathbb{R}$ be the real numbers, and let $(\mathbb{R},d_{\mathbb{R}})$ be the real line. Let $f:W\to\mathbb{R}$ be of class $C^1$ on $W$ (via clause 3 there, $f$ being real-valued), with partial derivatives $\partial_i f$ in the sense of Partial Derivative on a Euclidean Open Set. Let $x=(x_1,\dots,x_n)\in\mathbb{R}^n$ and $h=(h_1,\dots,h_n)\in\mathbb{R}^n$, and for $\tau\in\mathbb{R}$ write $x+\tau h$ for the point of $\mathbb{R}^n$ whose $k$th coordinate is $x_k+\tau h_k$ for $k\in\{1,\dots,n\}$.

Then the following hold.

1. (Interval neighborhood of a segment) If $x+\tau h\in W$ for every $\tau\in\mathbb{R}$ with $0\le\tau\le1$, then there exists a real $r>0$ such that $x+\tau h\in W$ for every $\tau\in\mathbb{R}$ with $-r<\tau<1+r$.

2. (First derivative) Let $J\subseteq\mathbb{R}$ be an interval with $x+\tau h\in W$ for every $\tau\in J$, and let $F:J\to\mathbb{R}$ be given by $F(\tau)=f(x+\tau h)$. Then $F$ is differentiable at every interior point $\tau_0$ of $J$, with

3. (Continuity) With $J$ and $F$ as in claim 2, the function $F$ and, for each $i\in\{1,\dots,n\}$, the function from $J$ to $\mathbb{R}$ given by $\tau\mapsto \partial_i f(x+\tau h)$ are continuous at every point of $J$ relative to $J$, regarded as maps from the subset $J$ of $(\mathbb{R},d_{\mathbb{R}})$ into $(\mathbb{R},d_{\mathbb{R}})$.

4. (Second derivative) With $J$ as in claim 2, assume in addition that $f$ is of class $C^2$ on $W$, and let $G:J\to\mathbb{R}$ be given by $G(\tau)=\sum_{i=1}^n \partial_i f(x+\tau h)\,h_i$. Then $G$ is differentiable at every interior point $\tau_0$ of $J$, with

where $\partial_j\partial_i f$ is the iterated partial derivative of clause 4 of C^k Maps on a Euclidean Open Set.