real analysis - Proof that the cardinality of continuous functions on $mathbb{R}$ is equal to the cardinality of $mathbb{R}$.

Proof that the cardinality of continuous functions on $\mathbb{R}$ is equal to the cardinality of $\mathbb{R}$.

I think is should be proved with the help of Cantor-Bernstein theorem.
It is easy to show that cardinality of of functions continuous on $\mathbb{R}$ is at least continuum - this set contains constant functions and there is natural bijection between them and $\mathbb{R}$. So we have one injection for the Cantor-Bernstein theorem. Can you please help with another? Or maybe there is another way, without Cantor-Bernstein theorem?

Answer

As David Mitra said, a continuous function is completely determined by its restriction to $\Bbb{Q}$. Hence, the following map is injective

$$

\Gamma : C(\Bbb{R}) \to \Bbb{R}^\Bbb{Q}, f \mapsto f|_\Bbb{Q}.
$$

This implies

$$
|C(\Bbb{R})|\leq |\Bbb{R}^\Bbb{Q}|.
$$

I will let you take it from here.

Hint:

$|\Bbb{R}|=2^{|\Bbb{N}|}$