Let $f$ be an entire function. Consider $A=\{z \in \Bbb{C} : f^{(n)}(z)=0\; \text{for some}\; n \in \Bbb{N}\}$. Then how to prove if $A=\Bbb{C}$, then $f$ is a polynomial ?
This is same as proving if $f$ is not a polynomial then $A$ is not all of $\Bbb{C}$.
I show the above statement with a particular example, like $f(z)=\sin z$
How to prove generally ? Any ideas ?
Suppose $A=\mathbb{C}$, we want to show that $f$ is a polynomial. Let $X_n=\{x\in \mathbb{C}: f^{(n)}(x)=0\}$. If we know one of the $X_n$ contains a open ball then by the property of entire function $f^{(n)}=0$ everywhere. Now assume $X_n$ are nowhere dense so $\mathbb{C}=\cup_{i=0}^{\infty}X_i$ but that's not possible by Baire category theorem.