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 Quote by micromass The key to the upper bound is the function $$\mathcal{C}(\mathbb{R},\mathbb{R})\rightarrow \mathbb{R}^\mathbb{Q}:f\rightarrow f\vert_\mathbb{Q}$$ it suffices to show that this function is an injection (since $\mathbb{R}^\mathbb{Q}$ has cardinality c). For this, we must show that if $f\vert_\mathbb{Q}=g\vert_\mathbb{Q}$, then f=g. This holds since $\mathbb{Q}$ is countable.
Perhaps you can just help me with some of the terminology... I'm sorry but I simply don't understand what a lot of the symbols you used are. I've never encountered functions written with a set as a superscript or a subscript. Does the "C" mean cardinality? Also, I'm assuming "dense" is equivalent to "countable"