= Solution
Since $v=t+r_*$ and $dr_*=f^{-1}dr$,
$$
dt=dv-f^{-1}dr.
$$
Substitution into the metric gives
$$
\begin{aligned}
ds^2
&=-f(dv-f^{-1}dr)^2+f^{-1}dr^2+r^2d\Omega^2\\
&=\boxed{-f\,dv^2+2\,dv\,dr+r^2d\Omega^2}.
\end{aligned}
$$
These are <Ingoing Eddington-Finkelstein coordinates>. Every coefficient is finite at $r=2M$, and the determinant of the $v,r$ block is $-1$. Thus the apparent singularity there in Schwarzschild coordinates is a coordinate singularity; the metric extends smoothly across the <Schwarzschild event horizon>.
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