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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?

Alex Becker
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user116357
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  • I am fairly certain that this question was asked before, can't seem to find a link at the moment though. – Asaf Karagila Dec 17 '13 at 12:43
  • The answer below will give you half of what you need. For the other half, construct of set with cardinality of $\Bbb R$ of everywhere differentiable functions. – David Mitra Dec 17 '13 at 13:07

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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).

universalset
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  • To extend to this: http://math.stackexchange.com/questions/17914/cardinality-of-the-set-of-all-real-functions-of-real-variable/17915#17915 – DSquare Dec 17 '13 at 12:23