Iterative application of continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $\lim_{n \to \infty}f^{n}(x)$ exists $\forall x$.

The questionLet $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $\lim_{n \to \infty}f^{n}(x)$ exists $\forall x$. Define $S = \{\lim_{n \to \infty}f^{n}(x): x \in \mathbb{R}\}$ and $T = \{x \in \mathbb{R}: f(x)=x\}$. Then conclude what can be said about the inclusion relation between $S$ and $T$, either $S$ is properly contained in $T$ or vice versa, or $S=T$.

Thoughts

I somehow feel that the answer should be $S$ is contained in $T$, as the sequence will converge to a stable point, and not every stable point needs to be an attracting point. Though I am not sure how to prove this. When the function is contractive, there is Banach fixed point theorem to help, but when it's just continuous I don't know how to conclude anything concretely.