# Holomorphic function and nth derivative.

Let $K$ be a open connected subset of complex numbers and $f$ holomorphic on $K$. If $f=0$ on some open disc $D$ in $K$, then is it true that

1. $n$th derivative of $f$ is $0$ for all points in $D$

2. And this implies $f=0$ on the whole of $K$.

### Progress

I know that holomorphic implies power series expansion exists about the point and was trying to see if it can be done using just this. My main question is, if $f$ is holomorphic and zero in some disc then is it zero on any connected set containing that disc?

• Please add some context or thoughts of your own. For example: do you know the identity theorem for holomorphic functions? – mrf Dec 16 '14 at 11:44
• well i know that holomorphic implies power series expansion exists about the point and was trying to see if it can be done using just this. – mac Dec 16 '14 at 11:48
• My main question is, If f is holomorpic and zero in some disc then is it zero on any connected set containing that disc. – mac Dec 16 '14 at 11:52
• That is basically the identity principle. One way to proceed is to prove that the set where $f=0$ is both open and closed. – mrf Dec 16 '14 at 11:54
• Can you show me please how you show that the set where f=0 is both open and closed – mac Dec 16 '14 at 11:57