I have to prove that for every $ \alpha$ in $[0, \Omega)$, there exists a continuous function $f$ from $[0, \Omega)$ to $\mathbb R$ such that the pre-image of $0$ is $\{α\}$.
I really have no idea how to start and would really appreciate just a hint, please don't post the complete answer.
Thanks.
HINT: If $\alpha=0$ or $\alpha=\beta+1$ for some $\beta<\Omega$, then $\alpha$ is an isolated point, and you can actually choose $f$ so that its range is $\{0,1\}$. Otherwise, let $\langle\alpha_n:n\in\Bbb N\rangle$ be a strictly increasing sequence with limit $\alpha$, and consider making $f$ constant on the clopen intervals $(\alpha_n,\alpha_{n+1}]$.
Added: For example, have $f$ send the entire interval $(\alpha_n,\alpha_{n+1}]$ to $-2^{-n}$. To simplify matters, you can send $[0,\alpha_0]$ as well as $(\alpha_0,\alpha_1]$ to $-1$. Of course $f(\alpha)=0$, so you have only to decide where to send the clopen set $(\alpha,\Omega)$.