proof of the Cauchy integral formula 
$ D=D_{r}(c), r > 0 .$ Show that if $f$ is continuous in $\overline{D}$ and holomorphic in $D$, then for all $z\in D$: $$f(z)=\frac{1}{2\pi i} \int_{\partial D}\frac{f(\zeta)}{\zeta - z} d\zeta$$ 

I don't understand this question because I don't see how it is different to the special case of the Cauchy Integral formula. I would be very glad if somebody could tell me what the difference is and how to show that it is true. 

I am not sure what the "usual Cauchy integral formula" is for you but I am assuming it is the following result: 
Theorem If $f$ is holomorphic in an open set containing the closure of $D_{r}(c)=\{z\in\mathbb{C}:\left|z-c\right|<r\}$ for some $r>0$ and $c\in\mathbb{C}$, then $f(z)=\frac{1}{2\pi i}\int_{\left|\zeta-c\right|=r} \frac{f(\zeta)}{\zeta - z}d\zeta$ for all $z\in D_{r}(c)$. 
Note that the hypothesis of the theorem is that $f$ is holomorphic in an open set containing the closure of $D_{r}(c)$ (in particular, there is a smoothness condition on the restriction of $f$ to the boundary of $D_{r}(c)$), whereas in the result stated in your question, we are only given continuity on the closure of $D_{r}(c)$.
In any case, the general result stated in your question is easy to prove. I will give a hint: fix $z\in D_{r}(c)$ and apply the usual Cauchy integral formula (i.e., the theorem stated above) to conclude that:
$f(z)=\int_{\left|\zeta-c\right|=\epsilon} \frac{f(\zeta)}{\zeta-z}d\zeta$
for all $\epsilon < r$ sufficiently close to $r$. You can now take the limit $\epsilon\to r$ in the above equality and apply the Lebesgue dominated convergence theorem to conclude the proof of the general result. (Please do not forget to verify that the hypothesis of the dominated convergence theorem is satisfied!)
I hope this helps!
