Problem

Let a\gt 0 be a real number and f(x) a real function defined on all of \mathbb{R}, satisfying for all x\in\mathbb{R},</p>

f(x)=\frac{1}{2}+\sqrt{f(x)-f^2(x)}.

a) Prove that the function f is periodic; i.e., there exists b\gt 0 such that for all x, f(x+b)=f(x).

b) Give an example of such a nonconstant function for a=1.

I placed a solution on a separate page.