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(If you have the text)pg 324 of Munkres topology. If $F$ and $F'$ are path homotopies between $f$ and $f'$ & $f'$ and $f''$, respectively, Munkres defines a path homotopy

$$ G(x,t)= \left\{ \begin{array}{cc} F(x,2t)\Huge\strut & t \in [0, 1/2] \\ F'(x,2t-1)\Huge\strut & t\in[1/2,1] \end{array}\right.$$

My problem is not in showing that $G$ is a homotopy, but that $G$ preserves the endpoints of the paths. Suppose that $f(0)=a$, $f(1)=b=f'(0)$, and $f'(1)=c$. Thus, we want $G(0,t)= a$ and $G(1,t)=c$. BUT how ho we know this happens since the value of $G$ changes with respect to $t$?? I.e how do we know both endpoints won't be the same?

Thanks in advance

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Nevermind, I falsely assumed there are three endpoints, when there are only 2. –  The Substitute Oct 20 '11 at 19:33

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I'm pretty sure your comment covers everything that I'm about to say - for the future, you can answer your own question and then accept that answer, I believe. But anyway:

If $f$ and $f'$ (or $f'$ and $f''$) are path-homotopic, then the two paths have the same initial and final endpoint. So we already know that $f(0) = f'(0)$ and $f'(0) = f''(0)$, and similarly $f(1) = f'(1) = f''(1)$.

We also know that if $F$ and $F'$ are the respective path homotopies, they each preserve the initial and final points (by definition of path homotopies). As a result, G also preserves the endpoints by its construction from $F$ and $F'$, which you seem to have a strong handle on already.

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