$f (f(x))=-x$ - there's no such function Prove that there's no continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(f(x))=-x $. 

Any hints?
Hint: $f \circ f \circ f=f^{-1}$ so the function is bijective, since the inverse exists. Every continuous bijection $\mathbb R \to \mathbb R$ is monotone.
How about $f(x)=ix$. Then $f(ix)=i(ix)=i^2x=-x$.