As shown here, the set of the continuous real functions $(ℝ,ℝ)$ has the same cardinality as the set of the real numbers $ℝ$.
Considering that it is easy to define a well-behaved distance in the set $(ℝ,ℝ)$, the immediate question would be: can you construct a continuous mapping $Φ: ℝ → (ℝ,ℝ)$ that is also surjective?
Since all the functions would be continuous, they'd be completely determined by their restriction to the rational numbers $Φ: ℚ → (ℚ,ℝ)$; by playing around with pairs of countable indices, it is easy to approximate a given continuous function - I wonder if there is a diagonal argument that can be used to prove or disprove the existence of a continuous mapping that can approximate any continuous function.