Let $f(z)$ be continuous in the closed disc Let $f(z)$ be continuous in the closed disc $\overline{D}(a,R)$ and analytic in the disc $D(a,R)$. Show that $$f(a)=\frac{1}{2 \pi}\int_0^{2\pi}f(a+Re^{I \theta})d\theta$$ 
More Generally, show that $$f^{(n)}(a)=\frac{n!}{2 \pi R^n}\int_0^{2\pi}f(a+Re^{I \theta})e^{-in\theta}d\theta \qquad  (n\ge 0)$$
I have been working on this problem for a pretty long time now I am not seeing how I can incorporate $f(z)$ being closed in one disc but analytic in another.

Hint: This is true for any $r, 0 < r<R.$ Consider the limit of this as $r\to R.$
