Entire function, periodic on real line implies periodic on $\mathbb{C}$?

Question

I'm preparing for an entrance exam and I came accross this Complex analysis problem: Let $g$ be an entire funciton so that $g(x)=g(x+1)$ for every real $x$. Is it necessarily true that $g(z)=g(z+1)$ for every complex $z$?

I am not sure how to proceed on this question. I know of a result that if an entire function is periodic, then it is of the form $f(z)=\sum_{n\in \mathbb{Z}}c_ne^{niaz}$. I am not sure if this is helpful. Thanks for the help.

Answer 1

Hint #1: Examine the function $f(z)=g(z+1)-g(z)$.

Mouseover to see hint #2:

Any holomorphic function which vanishes on the whole real line is identically zero.

Answer 2

Hint: Use the identity theorem for holomorphic functions, stating that if $f$ and $g$ are holomorphic on a domain $D$, and $f=g$ on a set $S\subset D$ with an accumulation point, then $f=g$ on $D$.