= Solution
Work on a connected regular radial interval, where $f$ is smooth and nonzero; replacing $f$ by $|f|$ allows us to take $f>0$. The only nonzero <Christoffel symbols> are
$$
\Gamma^r{}_{\phi\phi}=-ff',\qquad \Gamma^\phi{}_{r\phi}=\Gamma^\phi{}_{\phi r}=f'/f.
$$
Along the circle, write a <tangent vector> in the single-valued <orthonormal basis> $E_r=\partial_r$, $E_\phi=f^{-1}\partial_\phi$ as $V=uE_r+vE_\phi$. The equations for <parallel transport> reduce to
$$
\frac{du}{d\phi}=f'(r_0)v,\qquad \frac{dv}{d\phi}=-f'(r_0)u.
$$
Consequently $u+iv$ acquires the factor $e^{-2\pi i f'(r_0)}$ after one circuit. The basis itself agrees at $\phi=0$ and $2\pi$, so no extra basis change is missing. The <holonomy> is the identity on every <tangent vector> precisely when
$$
\boxed{f(r_0)\ne0,\qquad f'(r_0)\in\mathbb Z.}
$$
The first condition makes this a regular circle of the <Riemannian metric>; the second is the actual transport condition. In particular, zero slope is sufficient but not necessary. This is <holonomy around circular fibres>.
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