RO-NMO 2005 Problem 112

Let $f:[0,1)\to (0,1)$ a continous onto (surjective) function.

a) Prove that, for all $a\in(0,1)$ , the function $f_a:(a,1)\to (0,1)$ , given by $f_a(x) = f(x)$ , for all $x\in(a,1)$ is onto; b) Give an example of such a function.