Does there exist a function such that

[epkid08]

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

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

 

[CRGreathouse]

Is that \exists g\exists F, \forall g\exists F, or \exists F\forall g?

 

[epkid08]

\exists F\forall g

 

[willem2]

suppose g_0 = 1, g_1 = 2 and g_2 = 1

Then f(2) - f(1) = \frac {g_2 - g_1} {g_1} = 1 and

f(1) - f(2) = \frac {g_3 - g_2} {g_2} = -1/2

so there can't be any F for this g

 

[epkid08]

Can you generalize a non-piecewise function for g that has the values g_0 = 1, g_1 = 2, and g_2 = 1?

 

[willem2]

Can you generalize a non-piecewise function for g that has the values g_0 = 1, g_1 = 2, and g_2 = 1?

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

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

 

[epkid08]

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

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