Let $f: R \rightarrow R$ be a continuous function such that $ \lim_{n \to \infty } f^n(x) $ exists for every $x \in R $, where $f^n(x)=f \circ f^{n-1}(x) $ for $n \ge 2.$ Define $$ S=\{\lim_{n \to \infty }f^n(x): x\in R\}, T =\{x\in R:f(x)=x\}$$ Then which of the following is necessarily true?
$(A) S \subset T$
$(B) T \subset S$
$(C) S = T$
(D) None of the above
My effort:
Case 1: Taking $f(x) = k,$ any constant.
$f^2(x) = f\circ f(k)= f(k) =k$ repeating this process, we get $f^n(x) = k$, so $S=\{k\}$. $f(x)=x$ will be true only for x=k, so $T=\{k\}$. Hence S=T.
Case 2: Taking $f(x) = x^2.
f^n(x) = x^{2n}, S=\{\infty \}$.
$f(x) =x \Rightarrow x^2 - x = 0 \Rightarrow x = 0, 1.$ $ T = \{0,1\}. S\ne T.$
So option D is correct. Can someone please suggest if my answer is correct? If not, please provide reason with example.