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It seems to me that the first definition given is just strictly stronger than the second definition. I've tried to construct a function, using definition 2, that satisfies definition 1 but haven't made any noteworthy progress. Urysohn's lemma doesn't seem helpful here either.

Apparently, there's a really simple solution that only requires elementary facts about continuous functions but I haven't found anything useful over the span of multiple hours :P

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Find a function $f_1$ for $C$, $f_2$ for $D$ and consider $h=\frac{f_1}{f_1+f_2}$. As $f_1$ and $f_2$ are never $0$ at the same time (and have values in $[0,1]$, so are $\ge 0$) this is well-defined, and continuous, as division is continuous on $\Bbb R$. Note that:

$f^{-1}[\{1\}]= f_2^{-1}[\{0\}$ and $f^{-1}[\{0\}] = f_1^{-1}[\{0\}]$ etc.

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