I am studying for a real analysis final tomorrow and I stumbled across this interesting problem that I am wondering how to do. It goes as follows:
"Suppose that the function $f$ is analytic on $(a,b)$, prove $f(x) = 0$, $\forall x \in (c,d) $ $\subset (a,b) \implies f(x) = 0, \forall$ $x \in (a,b)$."
I think that using induction is the way to go. We know that the infinite derivatives on the interval $(c,d)$ are all $0$. How do we use this to find out that all the infinite derivatives on $(a,b)$ are all identically $0$?