2

In his book, Reverse Mathematics, John Stillwell says (pg. 44) "that each continuous function on R may be encoded by a set of natural numbers and hence the arithmetization project extends at least as far as the continuous functions. This remarkable result is due to Borel (1898), p. 109, and it follows that each continuous function may be encoded by a real number."

The referenced book by Borel is in French.

I can follow that a continuous function on R may be encoded by a set of natural numbers. But how does one go from that set to a real number specifically?

Of course I tried to search for this result on the innertube but, alas, I got nowhere. I'm not even sure what category in mathematics to start with to narrow the search down.

Also, given the real number, is it possible to reverse the encoding to get the continuous function back? Does every real number represent a continuous function? What book(s) are available to learn more about this intriguing result?

vahed
  • 21
  • Are you trying to construct a bijection between the reals and all continuous functions? – kam May 20 '20 at 19:08
  • https://math.stackexchange.com/questions/2127258/why-is-the-set-of-all-continuous-functions-of-size-beth-one – kam May 20 '20 at 19:49
  • No, I'm not trying to construct a bijection between reals and all continuous functions. I'm just trying to understand Stillwell's text more deeply. If anything, I would like to understand the encoding that Stillwell alludes to. I think an encoding is different than a bijection. An encoding might imply that decoding is possible in some algorithmic way, where the function can actually be reconstructed (or evaluated) given the real number. – vahed May 22 '20 at 02:32
  • @vahed I wouldn't be so sure that by "encoded" he means a computable bijection. I would say encoded just by the existence of the bijection. It is not a practical statement then. Just theory. – Jens Renders May 22 '20 at 14:55

1 Answers1

0

I think it is trying to say there is a bijection between $C^0(\mathbb{R}),$ the set of continuous functions on $\mathbb{R}$, and $\mathbb{R}$. In other words, $C^0(\mathbb{R})$ is uncountable.

edit:

We know due to the Weierstrass Approximation Theorem that the set of polynomials is dense in the set of continuous functions (both on the reals).

kam
  • 1,255
  • Maybe I should add, and to further quote Stillwell. "Is it not surprising that each continuous function, no matter how complicated, can be captured by a point on the number line?" – vahed May 20 '20 at 18:35
  • showing such a bijection exists and constructing such a bijection are very different things. – kam May 20 '20 at 18:41
  • I don't agree with "In other words, $C^0(\mathbb{R})$ is uncountable." That it is uncountable is trivial. That it doesn't have more elements than $\mathbb{R}$ is interesting. – Jens Renders May 20 '20 at 19:39