# Math Help - f(x)=x

1. ## f(x)=x

iIf $f:[0,1] \to [0,1]$ is an injection which satisfies $f(2x-f(x))=x$ then $f(x)=x$ for $x \in [0,1]$

2. Originally Posted by Chandru1
If $f:[0,1] \to [0,1]$ is an injection which satisfies $f(2x-f(x))=x$ then $f(x)=x$ for $x \in [0,1]$
Proof by contradiction: Suppose that $f(x)\ne x$ for some x, say $f(x) = x+\delta$ where $\delta\ne0.$ Then $x = f(2x-f(x)) = f(x-(f(x)-x)) = f(x-\delta)$, so that $f(x-\delta) = (x-\delta) + \delta$.

Now repeat the same argument with $x-\delta$ in place of x, to see inductively that $f(x-n\delta) = (x-n\delta) + \delta$ for n=1,2,3,... . But for n large enough, $x-n\delta$ will be negative (if $\delta>0$), or greater than 1 (if $\delta<0$). In either case that gives a contradiction.