UA-UMO 1998 Problem 114

Consider a function $f:[0,1]\to[0,1]$ . Suppose that there is a real number $0<\lambda<1$ such that $f(\lambda)\notin\{0,\lambda\}$ and the equality $$f(f(x)+y)=f(x)+f(y)$$ holds whenever the function is defined on the arguments.

(a) Give an example of such a function. (b) Prove that for some $x\in[0,1]$ , $$\underbrace{f(f(\cdots f(}_{19}x)\cdots))=\underbrace{f(f(\cdots f(}_{98}x)\cdots)).$$