The Line t ↦ x+th and Values of a Function along a Line in ℝⁿ, as Functions

In the setting of Euclidean Space and Real Matrices: Standing Notation, let $n\in\mathbb{N}$, $x,h\in\mathbb{R}^{n}$ and $J\subseteq\mathbb{R}$, with the operations of Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iv); maps and their values are those of Functions, Values of a Function, and Functions from One Class to Another (iii) and Functions, Values of a Function, and Functions from One Class to Another (ii).

(i) The line as a map. There is a map $\gamma:J\to\mathbb{R}^{n}$ with

$$\gamma(t)=x+th\qquad\text{for every }t\in J.$$

(ii) Values along a line. Let $U\subseteq\mathbb{R}^{n}$, let $Y$ be a class and let $F:U\to Y$ be a map with $x+th\in U$ for every $t\in J$. There is a map $\varphi:J\to Y$ with

$$\varphi(t)=F(x+th)\qquad\text{for every }t\in J.$$