The key to the upper bound is the function
[tex]\mathcal{C}(\mathbb{R},\mathbb{R})\rightarrow \mathbb{R}^\mathbb{Q}:f\rightarrow f\vert_\mathbb{Q}[/tex]
it suffices to show that this function is an injection (since [itex]\mathbb{R}^\mathbb{Q}[/itex] has cardinality c). For this, we must show that if [itex]f\vert_\mathbb{Q}=g\vert_\mathbb{Q}[/itex], then f=g. This holds since [itex]\mathbb{Q}[/itex] is countable.
