It is straightforward to see that $f \circ f$ is odd whenever $f$ is odd. Indeed, assuming $f(-x) = -f(x)$ for all $x$, we get
$$ f(f(-x)) = f(-f(x)) = -f(f(x)). $$
Hence, $f \circ f$ is an odd function as well.
My question is a converse of the above statement.
Suppose $f : \mathbb{R} \to \mathbb{R}$ is continuous. If $f \circ f$ is an odd function. What can I say about $f$ itself? Is it odd?