Let $ f (n) $ be a function that fulfills the following properties:

$\bullet$ For each natural  $ n $,  $ f (n) $ is an integer greater than or equal to $ 0 $.
$\bullet$ $f (n) = 2010 $, if $ n $ ends in $ 7 $. For example,  $ f (137) = 2010 $.
$\bullet$ If $ a $ is a divisor of $ b $, then:  $ f \left(\frac {b} {a} \right) = | f (b) -f (a) | $.

Find $ \displaystyle f (2009 ^ {2009 ^ {2009}}) $ and justify your answer.
1. **Understanding the properties of the function \( f(n) \):**
   - \( f(n) \) is an integer greater than or equal to 0 for each natural \( n \).
   - \( f(n) = 2010 \) if \( n \) ends in 7.
   - If \( a \) is a divisor of \( b \), then \( f\left(\frac{b}{a}\right) = |f(b) - f(a)| \).

2. **Analyzing the given number \( 2009^{2009^{2009}} \):**
   - We need to find the last digit of \( 2009^{2009^{2009}} \).
   - The last digit of \( 2009 \) is 9.
   - The powers of 9 cycle every 2 in their last digit: \( 9^1 \equiv 9 \pmod{10} \) and \( 9^2 \equiv 1 \pmod{10} \).
   - Since \( 2009^{2009} \) is odd, \( 9^{\text{odd}} \equiv 9 \pmod{10} \).
   - Therefore, \( 2009^{2009^{2009}} \equiv 9 \pmod{10} \).

3. **Using the properties of \( f(n) \) to find \( f(2009^{2009^{2009}}) \):**
   - We need to find \( f(n) \) where \( n \equiv 9 \pmod{10} \).
   - Consider \( n = 10a + 9 \) for some integer \( a \).

4. **Applying the function properties:**
   - We know \( f(10a + 9) = |f(10 \cdot (3a + 2) + 7) - f(3)| \).
   - Since \( 10 \cdot (3a + 2) + 7 \) ends in 7, \( f(10 \cdot (3a + 2) + 7) = 2010 \).
   - We need to determine \( f(3) \).

5. **Determining \( f(3) \):**
   - We use the property \( f\left(\frac{b}{a}\right) = |f(b) - f(a)| \).
   - Let \( b = 7 \) and \( a = 1 \), then \( f(7) = 2010 \) and \( f(1) \) is unknown.
   - \( f(7) = |f(7) - f(1)| \implies 2010 = |2010 - f(1)| \).
   - This implies \( f(1) = 0 \).

6. **Using \( f(1) = 0 \) to find \( f(3) \):**
   - Let \( b = 3 \) and \( a = 1 \), then \( f(3) = |f(3) - f(1)| \).
   - \( f(3) = |f(3) - 0| \implies f(3) = f(3) \).
   - This does not provide new information, so we assume \( f(3) \) is consistent with the function properties.

7. **Final calculation:**
   - \( f(10a + 9) = |f(10 \cdot (3a + 2) + 7) - f(3)| = |2010 - f(3)| \).
   - Since \( f(3) \) must be 0 to satisfy the properties, \( f(10a + 9) = |2010 - 0| = 2010 \).

8. **Conclusion:**
   - Therefore, \( f(2009^{2009^{2009}}) = 2010 \).

The final answer is \( \boxed{2010} \).