Please look at page 178 in Hatcher's algebraic topology: I quote the following passage:
Both $f(x)$ and $g(x)$ lie in a single simplex of $L$ since $g(x)$ lies in $[g(v_1),\ldots,g(v_n)]$ and $f(x)$ lies in the star of this simplex. So taking the linear path $(1-t)f(x)+tg(x),\,0\leq t \leq 1$, in the simplex containing $f(x)$ and $g(x)$ defines a homotopy from $f$ to $g$. To check continuity of this homotopy it suffices to restrict to the simplex $[v_1,\ldots,v_n]$, where continuity is clear since $f(x)$ varies continuously in the star of $[g(v_1),\ldots,g(v_n)]$ and $g(x)$ varies continuously in $[g(v_1),\ldots,g(v_n)]$.
Could someone kindly explain the above paragraph in greater detail, especially the line beginning "To check continuity..."?
I have these questions:
Firstly, to be able to write $(1-t)f(x)+tg(x)$, we must first know why $f(x)$ and $g(x)$ are in a single simplex-- but this has been proved in the lines that precede the ones quoted above. For this homotopy to be continuous, we must know that (please refer to the proof in the text) the simplex of $L$ that contains $f(x)$ and $g(x)$, let us call it $\Delta_x$, should be same for every $x \in int[v_1,\ldots,v_n]$. I don't know how to see this.
Secondly, even assuming that $\Delta_x$ is same across $x \in int[v_1,\ldots,v_n]$, we can only conclude that the homotopy is continuous on $int[v_1,\ldots,v_n]$-- how to see that this homotopy is (1) well-defined and (2) extends continuously to $[v_1,\ldots,v_n]$ and in turn to $K$?
Thank you very much.