A sufficient condition for the existence of a fixed point for a continuous function.

Question

How did the author use the intermediate value theorem to prove that period $k$ implies period $1$?

Please, see the image which explains every thing. The definitions are in the first paragraph. The question is related to the second paragraph. .

Thanks in advance.

Answer 1

Let $g(x) = f(x) - x$, then $g(a) = f(a) - a > 0$ and $g(b) = f(b) - b < 0$, therefore since $g$ is continuous there must be a point between $a$ and $b$ such that $f(c) - c = g(c) = 0$, that is $f(c) = c$, a fixed point for $f$.