smooth immersion on the circle is periodic

Question

Consider a smooth immersion ($C^\infty$ with nonvanishing first derivative) $f:S^1\rightarrow\mathbb{R}^2$. Why is such a function necessarily periodic? This was stated in a paper ($1.4$ in https://arxiv.org/pdf/1807.11290.pdf). To me, for any such function $f$, we have something like $$f(\cos(x),\sin(x))=f(\cos(x+2\pi),\sin(x+2\pi))$$ so that $f$ is not periodic, but the function $f\circ\varphi:\mathbb{R}\rightarrow\mathbb{R}^2$, where $\varphi:\mathbb{R}\rightarrow S^1$ sends $x\mapsto (\cos(x),\sin(x)),$ is periodic. Maybe something like this is meant. But even this only seems to have meaning if we are letting $x$ range over all of $\mathbb{R}$. However, $S^1$ is parametrized just by $S^1=\{e^{ix}\in\mathbb{C}: x\in[0,2\pi)\}$, so the map above would actually just be $f\circ\varphi:[0,2\pi)\rightarrow\mathbb{R}^2$, which is not $2\pi$-periodic since $x+2\pi$ is not in the domain of $f\circ\varphi$ for $x\in[0,2\pi)$. It would be like saying $g:[0,2\pi)\rightarrow\mathbb{R}$ mapping $x\mapsto\cos(x)$ is periodic. So my question is why are smooth immersions on the unit circle necessarily periodic?