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?
Proof that a continuous function with $f(x) = f(x^2)$ is constant.
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2please add some of your thoughts and approaches and we will be happy to guide you further – 2017-01-06
1 Answers
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$f$ 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 $f(0)=f(1)$.