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To prove the statement, there is a function $f$ such that $f′ = f$, we show that $f(x) = \exp(x)$ works.

That is clear, so long as there is a function $f$ such that $f(x) = \exp(x)$. But how do we show that there is a function $f$ such that $f(x) = \exp(x)$?

[EDIT] To clarify this question, how also do we prove that there is a function $g$ such that $g(n) = n+n$?

[EDIT] Or even simpler, how do we prove that there is a function $h$ such that $h(n) = 1$, for any $n$?

[EDIT] By a function, I mean any relation that associates each element $x$ of a set $X$ to a single element $y$ of a set $Y$ (possibly the same set as $X$). How do we prove that such a relation exists?

amWhy
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buckner
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    If you have a definition of exp pointwise, e.g. Taylor expansion, then it exists as the limit. If you have a definition for Euler's number and powers, then you also have existence. I guess it depends on what you mean with "how do we know there is a function exp" – I was suspended for talking Feb 16 '20 at 11:46
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    From your question(s) it isn't clear to me what you mean by "function", and also which things you don't doubt to exist. (You can take a philosophical position that nothing exists, but then we won't have a lot of common ground for discussion.) –  Feb 16 '20 at 11:55
  • "how do we prove that there is a function h such that h(n) = 1, for any n?" Defining it : $h = { (n,1) \mid n \in \mathbb N }$ – Mauro ALLEGRANZA Feb 16 '20 at 12:11
  • Do you accept that sets, e.g. as described by Zermelo-Fraenkel axioms, exist? (https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory) If yes, we could seek to outline the proof within that theory. –  Feb 17 '20 at 12:23
  • Yes, why not assume that sets exist. From memory, one can define a function in terms of set theory, but 20 years since I last looked at this. (EDIT) Just found this https://math.stackexchange.com/questions/60365/what-is-the-set-theoretic-definition-of-a-function – buckner Feb 18 '20 at 12:21

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