Comment by Gro-Tsen on Is there an infinite subset of $\Bbb{R}$ not homeomorphic to any of its proper subsets?

I'm confused about the claim “then $f_α$ continuously extends to an injection on $\mathbb{R}\setminus Y_α$”: your assumption is that for each $x \not\in X_α \cup Y_α$ separately, $f_α$ continuously and injectively extends to $x$, but why is it so for all $x$ simultaneously? For “continuous”, I see why being able to extend to each $x$ separately means we can extend to all at once; but for “injective” I don't see the argument.