Show that if $f,g:\omega_1 \to \omega_1$ preserve order (monotone increasing) and are continuous ($f(\beta)=\sup\{f(\alpha) : \alpha<\beta\}$ for all limit ordinals $\beta<\omega_1$), then they coincide on a club set S, that is $f\restriction S = g \restriction S $.
Also, for how many functions can we generalize this statement? (for any finite number of functions seems logical if the statement is true, what about more)
HINTS: For the first question, let $\alpha_0<\omega_1$ be arbitrary. Given $\alpha_n$, let
$$\alpha_{n+1}=\begin{cases} f(\alpha_n),&\text{if }n\text{ is even}\\ g(\alpha_n),&\text{if }n\text{ is odd}\;. \end{cases}$$
Let $\alpha=\sup_n\alpha_n$. What can you say about $f(\alpha)$ and $g(\alpha)$?
For the second question, what do you know about the intersections of finitely many closed unbounded sets? What about countably many?