8.56 Let $f(x)=x^{2}+b x+c$. If the equation $f(x)=x$ has no real roots, then the equation $f(f(x))=x$
(A) has 4 real roots.
(B) has 2 real roots.
(C) has 1 real root.
(D) has no real roots.
(Chinese Jiangsu Province Junior High School Mathematics Competition, 1991)
［Solution］Assuming the equation $f(x)=x$ has no real roots, this means the line $y=x$ and the parabola $y=f(x)$ have no intersection points. Given that the coefficient of the quadratic term in $f(x)$ is positive, we know it opens upwards.

Thus, the graph of $y=f(x)$ lies above the line $y=x$. That is, for any real number $x$, we have $f(x)>x$.

Therefore, $f(f(x))>f(x)>x$, which means $f(f(x))=x$ has no real roots.
Hence, the correct choice is $(D)$.