Polygonal paths and polygonally connected subsets of $\mathbb{R}^n$

Let $A\subseteq\mathbb{R}^n$. A polygonal path in $A$ from $x$ to $y$ is a path $\gamma:[0,1]\to A$ (Paths, path-connected spaces and path components, Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) for which there are a finite list of vertices $v_0,\ldots,v_m\in A$ and a partition $0=t_0<t_1<\cdots<t_m=1$ such that $v_0=x$, $v_m=y$, and

$\gamma(t)=\frac{t_i-t}{t_i-t_{i-1}}v_{i-1}+\frac{t-t_{i-1}}{t_i-t_{i-1}}v_i\quad\text{when }t_{i-1}\le t\le t_i.$

The formula uses only scalar multiplication and vector addition in $\mathbb{R}^n$ (Vector space over a field). The finite list is indexed by a natural number (The cardinality $\lvert A\rvert$ of a finite set).

The subset $A$ is polygonally connected when every pair of its points is joined by a polygonal path in $A$.

A polygonal path is required to be a path, so its continuity is part of the definition. The next lemma verifies continuity for the displayed finite concatenations when their image lies in the stated subset.