# Does there exist a function such that

1. Nov 16, 2009

### 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

Last edited: Nov 16, 2009
2. Nov 16, 2009

### CRGreathouse

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

3. Nov 16, 2009

### epkid08

$$\exists F\forall g$$

Last edited: Nov 16, 2009
4. Nov 16, 2009

### 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

5. Nov 16, 2009

### epkid08

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

6. Nov 16, 2009

### willem2

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 $g_n$

7. Nov 17, 2009

### epkid08

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