elementary set theory - Set of continuous functions from $mathbb{R}$ to $mathbb{R}$ is equinumerous to $mathbb{R}$

I am trying to prove that the set $\mathcal{C}:=\{f\mid f\in\mathbb{R}^\mathbb{R}\mid f \text{ is continuous}\}$ is equinumerous to $\mathbb{R}$.

To achieve this, I note that an $f\in\mathcal C$ is ''pinned down'' by $\mathbb{Q}$, i.e. if I know how $f$ behaves on $\mathbb{Q}$, I know all of $f$, since $\mathbb{Q}$ is dense in $\mathbb{R}$ and $f$ is continuous. So there is an easy injection from $\mathcal{C}$ to $\{f\mid f\in\mathbb{R}^\mathbb{Q}\mid f\text{ is continuous}\}$.

The latter set can be embedded into $\mathbb{R}^\mathbb{N}$, but I'm afraid that fact won't help me, since (as I believe): not $\mathbb{R}^\mathbb{N}\preccurlyeq\mathbb{R}$.

Could someone hint me into the right direction on proving $\mathcal{C}\sim\mathbb{R}$ (via $\mathcal{C}\preccurlyeq\mathbb{R}$)?

Answer

You have $$\vert \mathbb R^{\mathbb N} \vert = \vert (2^{\mathbb N})^{\mathbb N} \vert = \vert 2^{\mathbb N \times \mathbb N} \vert = \vert 2^{\mathbb N} \vert = \vert \mathbb R \vert$$