Exist strictly increasing $f \colon \omega _1 \mapsto \mathbb{R}$

Question

Does there exist a strictly increasing injective function $f \colon \omega _1 \mapsto \mathbb{R}$, where $\omega _1$ denotes the first uncountable ordinal?

Answer 1

No. If so, there is some rational between $f(\alpha)$ and $f(\alpha+1)$ for each $\alpha\in\omega_1$, and so uncountably many rationals, as these rationals must be distinct.