Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $\lim _{n\rightarrow \infty} f^n(x)$ exists for every $x \in \mathbb{R}$, where $f^n(x) = f \circ f^{n-1}(x)$ for $n \geq 2$. Define
$$S=\left\{\lim _{n \rightarrow \infty} f^n(x): x \in \mathbb{R}\right\} \text{ and } T=\left\{x \in \mathbb{R}:f(x)=x\right\}$$
Then which of the following is necessarily true?
64.3k questions
77.9k answers
244k comments
80.0k users