Hi. I'm trying to understand this example from Munkres's book and there is a part that I can't understand what the definition of the $g$ loop is. More precisely, I don't understand "For each $n$, define $g$ on the interval $[1/(n+1),1/n]$ to be positive linear map of this interval onto $[0,1]$ followed by $f_n$"
What would the definition of $g$ look like? what value does the function $g$ take in that interval? I can't understand this part. Thanks.
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We define $g_n$ to be the function $g_n: [1/(n+1), 1/n] \to [0,1]$ defined by $$ g_n(x) = n(n+1)x-n $$ This is a line of positive slope connecting the points $(1/(n+1)),0)$ and $(1/n,1)$. By design, $g_n$ is bijective. We can then form the composition $f_n \circ g_n: [1/(n+1), 1/n] \to C_n$ (this is what he means by "followed by $f_n$"). Doing this for all $n$ and patching them together defines the loop $g$ everywhere but $0$, so he defines $g(0)=p$ to complete the loop continuously.
$$g(x) = \begin{cases} p & x = 0 \\ f_n\left(\frac{x-\frac{1}{n+1}}{\frac{1}{n} - \frac{1}{n+1}}\right) & x \in \left[\frac{1}{n+1}, \frac{1}{n}\right], n \ge 1 \end{cases}$$
The map $x \mapsto \frac{x-\frac{1}{n+1}}{\frac{1}{n} - \frac{1}{n+1}}$ is the "positive linear map" from $[1/(n+1), 1/n]$ that is surjective onto $[0,1]$.