Find all continuous functions $f:\mathbb{R}\longrightarrow \mathbb{R}$ so that $f(x) = f(x^2+{1\over 4})$ is valid for all $x$.
Well, I thought that I solved this one, but then one of my student noticed obvious mistake. Namely I defined the sequence $x_{n+1}= x^2_n +1/4$ and $x_0=a$ where $a$ is arbitrary real number. It is easy to see that it is increasing and it has lover bound. I was trying to arguing that it converge to $1/2$ and since $f(x_{n+1})=f(x_n)$ it has constant value on this sequence and so $f(a)=1/2$ is for all $a$. But this sequence is not convergent for all $a$. Any ideas how to repair this ''prove''.