Let $f:D\to D$, where $D\subseteq \mathbb{R}^n$, be a continuous function. Under what conditions is $f\circ f \circ \cdots$ continuous? Here, $\circ$ stands for the composition operator and sometimes the notation $f^2=f\circ f$ is used. So in this notation, when is $\lim_n f^n$ continuous?
There is a counterexample I can think of: $f(t)=t^\alpha$ over $D=[0,1]$ for $\alpha>1$. Then $f^n(t)=t^{\alpha^n}$ and the point-wise limit of $f^n$ is $\left(\lim_n f^n\right)(t)=0$ for $t\in[0,1)$ and $\left(\lim_n f^n\right)(1)=1$ which is not continuous.
One think one could suggest is that $f^n$ be a uniformly convergent sequence of functions. But what should this imply for $f$?