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 Blog Entries: 8 Recognitions: Gold Member Science Advisor Staff Emeritus 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.