Cardinality of the set of differentiable functions
Is the cardinality of $$X = \{f: \Bbb R \to \Bbb R \;|\; f \text{ is differentiable everywhere}\}$$ the same as $\Bbb R$? How to prove it?
Solution 1:
Hint: This is also true of the set of continuous functions from $\mathbb{R}$ to $\mathbb{R}$, which contains all differentiable functions. Try using the fact that a continuous function is determined by its values on the rational numbers (a countable set).