TITLE: Proving existence of at least one root
QUESTION [5 upvotes]: The function $f:\mathbb{R}\to\mathbb{R}$, is continuous and $a>0$.
How can I prove that there is at least one root of this equation:
$f(x)=f(\sqrt{|x^2-a|})$

REPLY [9 votes]: The function $x\mapsto f(\sqrt{|x^2-a|})-f(x)$ is continuous, maps $0\mapsto f(\sqrt a)-f(0)$ and $\sqrt a\mapsto f(0)-f(\sqrt a)$. Now use the IVT.