Given a continuous function $$ f: \mathbb{R} \rightarrow \mathbb{R} \quad \text{with} \quad f(x) = f(x^2) \quad\quad \forall x \in \mathbb{R} $$

How can I show that $f$ must be constant?

  • 2
    $\begingroup$ please add some of your thoughts and approaches and we will be happy to guide you further $\endgroup$ – gt6989b Jan 6 '17 at 16:06

$f$ is an even function. let $x> 0$ and $n\geq 0$. we have $$f(x^2)=f(x)=f(\sqrt{x})=$$ $$f(x^{\frac{1}{2^n}}).$$

by continuity of $f$,

when $ n\to +\infty, f(x)=f(1)$.

and $f(0)=f(1)$.


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