Prove fundamental group of R/~ is isomorphic to Z

Question

I need help with this $$\pi_1(\mathbb{R}/\sim)\simeq \mathbb{Z}$$ I am not sure if this proposition is true or false. Let $f:\mathbb{R}\to\mathbb{C}: f(x)=e^{2\pi ix}$ and let $\sim$ the equivalence relation defined: $x\sim y\Leftrightarrow f(x)=f(y)$. So I think that $\pi_1(\mathbb{R}/\sim)\simeq \mathbb{Z}$ is false but I don't know how to prove it. I know that $\mathbb{R}/\sim$ is path connected and also that $f$ is continous but I can't find a isomorphism or contradiction.