## The Function f(x) = C and the Continuum Hypothesis

**Authors:** Ron Ragusa

Part 1 examines whether or not an analysis of the behavior of the function f(x) = C, where C is any constant, on the interval (a, b) where a and b are real numbers and a < b, will provide a method of proving the truth or falsity of the Continuum Hypothesis (CH). The argument will be presented in three theorems and one corollary. The first theorem proves, by construction, the countability of the domain d of f(x) = C on the interval (a, b) where a and b are real numbers. The second theorem proves, by substitution, that the set of natural numbers ℕ has the same cardinality as the subset S of real numbers on the given interval. The corollary extends the proof of theorem 2 to show that ℕ and ℝ are of the same cardinality. The third theorem proves, by logical inference, that the CH is true.
Part 2 is a demonstration of how the set of natural numbers ℕ can be put into a one to one correspondence with the power set of natural numbers, P(ℕ). From this I will derive the bijective function f : ℕ → P(ℕ). Lastly, I’ll propose a conjecture asserting that f(x) = C can be employed to construct a one to one correspondence between the natural numbers and any infinite set that can be cast as the domain of the function.
Appendix A extends the methodology for creating a bijection between infinite sets to the function f(x) = x2 using random real numbers from the domain of the function as input to f(x) = x2 in order to show how the constructed array would appear in practical application.

**Comments:** 14 Pages. email: ron.ragusa@gmail.com

**Download:** **PDF**

### Submission history

[v1] 2018-06-03 07:41:10

[v2] 2018-06-04 08:33:05

[v3] 2018-06-27 22:04:44

**Unique-IP document downloads:** 2070 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary.
In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution.
Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

**Add your own feedback and questions here:**

*You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.*

*
*