Question 5
Let $a>0$ be a real number and $f(x)$ a real function defined on all of $\mathbb R$, satisfying for all $x\in\mathbb R$,
$$f(x+a)=\frac12+\sqrt{f(x)-f(x)^2}.$$
(a) Prove that the function $f$ is periodic; i.e., there exists $b>0$ such that for all $x$, $f(x+b)=f(x)$.
(b) Give an example of such a nonconstant function for $a=1$.
$$f(x+a)=\frac12+\sqrt{f(x)-f(x)^2}.$$
(a) Prove that the function $f$ is periodic; i.e., there exists $b>0$ such that for all $x$, $f(x+b)=f(x)$.
(b) Give an example of such a nonconstant function for $a=1$.