(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