I want to find a non-constant, continuous function $f: [2, \infty) \rightarrow \mathbb{R}$ which satisfies $f(x) = f(x^2)$ for all $x \in [2, \infty)$. A friend of mine suggested to try something of the form $f(x) = \sum_{n=-\infty}^\infty \varphi(2^n \log x)$ and we tried $\varphi(x) = \frac{x}{1+x^2}$. Now my friend plotted the function for $-1000 \leq n \leq 1000$ which gave the following result:
Now we let MATLAB compute $f(x)$ for a few $x \in \mathbb{N}$ and observed that $f(x) \approx 2.2662$. As the plot of the function did not imply that the function was constant but periodic (however with a very small amplitude), we computed a few other values, e.g. $x = \pi$, $x = \pi+1$ and we were surprised to see that again $f(x) \approx 2.2662$. For large values, MATLAB was no more capable of computing the result.
So I mainly have have two questions:
- Is this function constant? How can I see this and prove or disprove it?
- Is there a more trivial example of such a function which is non-constant?
Thanks for any answers in advance.