stretching [f(a), f(b)], f(a)<a, f(b)>b.
Now consider the function
This is negative at a, and positive at b, and continuous, so by
the intermediate value theorem there is a point c in [a,b] such that g(c)=0
so f(c)=c, that is there is a fixed of invariant point on the band.
(you could do this without constructing g, from f directly, but I prefer