0
votes
3answers
43 views

Rationals over an interval

Suppose $I$ is an interval $[a,b]$. It is noted that $a$ and $b$ are real integers. Divide the interval into $n$ parts with step size $h=(b-a)/n$. Clearly all the points $a$, $a+h$, ...
11
votes
1answer
105 views

Differentiation of a function $f:\mathbb{Q}\to \mathbb{Q}$(Rational Calculus)

Assume that $f:\mathbb{Q}\to \mathbb{Q}$ is given such that $\forall a\in \mathbb{Q}$ the following limit, exists \begin{equation} \lim_{x\to a} \frac{f(x)-f(a)}{x-a}\in \mathbb{R} ...
0
votes
3answers
57 views

the simplest function f(1)=-1; f(2)=0; f(3)=1; f(4)=0.

I'm looking for a function that gives f(1)=-1; f(2)=0; f(3)=1; f(4)=0. The other values are undefined and I don't pay any attention on them. The prefered ...
6
votes
4answers
298 views

Is there a bijection $f: \mathbb{Q} \to \mathbb{Q}_{>0}$?

For $\mathbb{R}$, we have the exponential function. Is there also a bijection $f: \mathbb{Q} \to \mathbb{Q}_{>0}$ or to $\mathbb{Q}_{\geq 0}$?
7
votes
2answers
220 views

Is there a non-constant smooth function mapping $\mathbb{R}$ into $\mathbb{Q}$

I cannot think of a non-constant smooth function which maps all real numbers into rational numbers. Can anyone give a simple example ? The simpler, the better !
0
votes
0answers
75 views

Limits as a representation of the Dirichlet function

I read that the Dirichlet function (1 if Rational, 0 else) can be written as: What is the proof of that? Are those limits commutative? Is there any other closed formula for Dirichlet function? (With ...
1
vote
1answer
34 views

Function of a rational number

Let a function exist such that $f(a+b)=f(a)+f(b)$. We have already shown that for any integer n, $f(nx)=n f(x)$. Now we must show that for any rational number $n/m$, $f(n/m)=n/m f(1)$. The problem is ...
0
votes
1answer
431 views

Inverse rational numbers counting function

I have found next formula of counting all rational numbers (or integer pairs, from which rational numbers can be builded): $N(i,j) = ((i - j - 1)(i - j) - 2j - 1)(1 + j / |j|)/2 + ((i + j - 1)(i + j) ...