Show that if $0\leq f( x) \leq 1$ for each $x\in [0,1] $
and $ f( x)$ is a continuous function, then there is some number $a$,
$0\leq a \leq 1$, such that $ f( a) = a$.
Tags: fixed-point, IVT, abstract-function, proof, tricks, classic, a--a, g t1, c3
I am always elated when a student draws the line y=x, the line x=1, the line x=0 and shows this graphically. I praise them for this and then ask them to prove it analytically. You may have to give them a hint to consider g(x)=f(x)-x on [0, 1] and use the IVT. Anymore than this hint will give the problem away.
– James Epperson 1998-08-09
This is a classic fixed point application of the IVT to f(x) - x. Once you’ve seen this trick, it becomes a doable problem, but its surprisingly hard without context.
– Eric Hsu 1998-08-09