Does there exist a function such that

**epkid08** · Nov16-09, 04:41 PM

If we define a finite difference operator as $LaTeX Code: \\Delta a_n = a_{n+1}-a_n$

Can we prove or disprove the existence of a function F, $LaTeX Code: F:\\mathbb{Z}\\rightarrow\\mathbb{Z}$ , such that $LaTeX Code: \\Delta F(g_n)=\\frac{\\Delta g_n}{ g_n}$ , where g is some arbitrary function?

Edit: fixed Big typo

**CRGreathouse** · Nov16-09, 05:02 PM

Is that $LaTeX Code: \\exists g\\exists F$ , $LaTeX Code: \\forall g\\exists F$ , or $LaTeX Code: \\exists F\\forall g$ ?

**epkid08** · Nov16-09, 05:35 PM

**willem2** · Nov16-09, 10:13 PM

suppose

and

Then $LaTeX Code: f(2) - f(1) = \\frac {g_2 - g_1} {g_1} = 1$ and

$LaTeX Code: f(1) - f(2) = \\frac {g_3 - g_2} {g_2} = -1/2$

so there can't be any F for this g

**epkid08** · Nov16-09, 11:21 PM

Can you generalize a non-piecewise function for g that has the values

, and

?

**willem2** · Nov16-09, 11:39 PM

Originally Posted by epkid08

Can you generalize a non-piecewise function for g that has the values , and ?

Why? Of course there is a quadratic going through these points.

Is the range of F really $LaTeX Code: \\mathbb{Z}$ ?. That is a problem with the division by

**epkid08** · Nov17-09, 10:16 PM

Originally Posted by willem2

Is the range of F really $LaTeX Code: \\mathbb{Z}$ ?. That is a problem with the division by

]

Now that I think about it, it shouldn't be.