Find the cardinality of the set of all continuous real-valued functions of one variable.

Question

Find the cardinality of the set of all continuous real-valued functions of one variable.

Trying to solve this problem I stumbled upon one casualty whichI've been struggling for a while with. This is my solution:

The cardinality of this set is bounded below by the cardinality of $\mathbb R$, because $f: \mathbb R \to \mathbb C(\mathbb R)$ given by $f(x) = \mbox{function y = x}$ is an injection. Now, I tried to bound this set from above:
$$|\mathbb R| = |\mathbb R^{\aleph_0}| = |\mathbb R^\mathbb Q|$$
And now I tried to make an injection $$I: \mathbb C(\mathbb R) \to \mathbb R^\mathbb Q$$
given by $$I(x) = x \upharpoonright \mathbb Q$$
However, in order for this solution to be acceptable, I'd have to prove this:
$$ x\upharpoonright \mathbb Q = y \upharpoonright \mathbb Q \Longrightarrow x=y$$
And this is where I got stuck. How can I tackle this proof?

$$\mathbb C(\mathbb R) - \mbox{Continuous real-valued functions}$$

Answer 1

$|C(\mathbb{R})|$ is at least $|\mathbb{R}|$ because for every $c \in \mathbb{R}$, the constant function $f(x) = c$ belongs to $|C(\mathbb{R})|$. It is at most $|\mathbb{R}^\mathbb{Q}|$, because $\mathbb{Q}$ is dense in $\mathbb{R}$ and so a continuous function on $\mathbb{R}$ is determined by its values on $\mathbb{Q}$. And as you note, $|\mathbb{R}| = |\mathbb{R}^\mathbb{Q}|$.