complex analysis - How do I show that an entire function which is real for all $zin mathbb C$ with $Im(z)=0$ or $pi$ is $2pi i$ periodic?

I have an entire function $f$ which takes real values on all complex numbers with imaginary parts $0$ or $\pi$. I'm trying to show that $f$ is periodic with period $2\pi$. I tried using the Cauchy-Riemann equations, but to no avail. How should I approach this problem?

Answer

As the entire function $\overline{f(\overline z)}-{f(z)}$ is $=0$ for $z\in\Bbb R$, this also holds for $z\in\Bbb C$.
Therefore, $f(a+i\pi)=\overline{f(a-i\pi)}=f(a-i\pi)$.
But then $f(z-\pi i)-f(z+\pi i)$ is entire and $=0$ on $\Bbb R$, hence on $\Bbb C$.