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 $v_0,\dots ,v_m\in A$ and a partition $0=t_0<t_1<\cdots <t_m=1$ such that $v_0=x$, $v_m=y$, and

$\gamma (t)=\frac{t_i-t}{t_i-t_{i-1}}{v}_{i-1}+\frac{t-t_{i-1}}{t_i-t_{i-1}}{v}_i\text{when }{t}_{i-1}\le t\le t_i\mathrm{.}$

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 $\mid A\mid$ 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.