does $f(x)$ defined by $f(x)=f(x^2)$ must be a constant? I proved that if $f(x)$ is continuous then it must be a constant. 
But I can't disprove or prove it for non-continuous function.

Similar to Robert's answer but perhaps simpler
$$ 
f(x) = \begin{cases} 1  \quad x \in [-1,1]\\ 0 \quad\text{ else}\end{cases}.
$$
