The existence of a continuous function with a given property I could use some help with the following problem
If $ a_0, a_1, \ldots , a_n $ are real numbers such that the equation 
$ a_nx^n+a_{n-1}x^{n-1}+ \cdots + a_0=0 $ has no real solutions, then there exists no continuous function  $ f:\Bbb R \mapsto \Bbb R $ such that 
$ a_nf^{[n]}+a_{n-1}f^{[n-1]}+ \cdots + a_0f^{[0]}=0 $, where $f^{[n]} $ is the $n$th iterate of $f$ (e.g $ f^{[0]}=1_{\Bbb R}, f^{[2]}=f \circ f$)
An example would be that there are no continuous functions $f$ such that $f(f(x))+f(x)+x=0$ , since the polynomial $x^2+x+1=0$ doesn't have real roots.(proof goes via injectivity)
I'm primarily looking for an elementary solution, but any one will do.
 A: you can prove that $$f(x)\neq x\; for\; x\neq0\;,\;\exists\lim_{x->+\infty }f(x)\in \left \{ +\infty,-\infty \right \}and\;f\;is\;strict\;Increasing\;or\;decreasing\;$$but I don't know is useful
A: Edit': Thanks to @Hurkyl I realized the combination of iterated functions is required to be constantly zero. My bad.
Edit: the assertion is false. $x^2+x+1$ has no root in $\mathbb{R}$, but setting $f(x)=-3x+1$ it is clear that $f^{[2]}+f^{[1]}+f^{[0]}$ has $x=0$ as zero. 
If you meant the identity function by $1_\mathbb{R}$, the above still holds.
I actually think it is always possible to find the linear combination of the coefficients I mentioned below and a corresponding continuous function since your polynomial equation is of a finite degree.
(Old answer)
Not an answer, just an idea.
Assume one can find distinct reals $c_1, c_2, \cdots, c_n$ such that $\sum_{k=1}^n a_k c_k =-a_0$. Then there must be a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ that passes through the points(with distinct horizontal positions) {$(c_0,c_1),\cdots,(c_{n-1},c_n)$}. Now that $\sum_{k=0}^n a_k f^{[k]} $ has $c_0$ as zero, and conversely if we can show that any continuous function meeting the requirements also has to pass through $n$ points with the above property, it will remain to show that it is impossible for the coefficients of a real polynomial with no real root, namely $a_1, a_2, \cdots, a_n$, to be linearly combined on $\mathbb{R}$ to yield $-a_0$.
But it is hard to characterize those coefficients so I'm going nowhere from here. On a side note, $n$ must be an even number, obviously.
