general relativity - Discontinuity of metric derivatives in the Israel junction formalism

It is often said that given the metrics $g^+$, $g^-$ on two sides of a hypersurface $\Sigma$, then, with a level-set function $\phi$ such that $\Sigma = \phi^{-1}(0)$, we can describe the metric on the whole manifold by

$$\begin{equation} g = \theta(\phi) g^+ + (1 - \theta(\phi)) g^- \tag{1} \end{equation}$$

And then, the derivatives of the components are simply

$$\begin{equation} g_{ab,c} = \partial_c \theta(\phi) (g^+ - g^-) + \theta(\phi) g^+_{ab,c} + (1 - \theta(\phi)) g^-_{ab,c}\tag{2} \end{equation}$$

and since it is assumed that $g$ is continuous,

$$\begin{equation} g_{ab,c} = \theta(\phi) g^+_{ab,c} + (1 - \theta(\phi)) g^-_{ab,c}\tag{3} \end{equation}$$

The discontinuity in the derivatives is then said to be

$$\begin{equation} [g_{ab,c}] = \gamma_{ab} n_c\tag{4} \end{equation}$$

for $n$ a normal form to $\Sigma$ and $\gamma_{ab}$ some tensor, and the notation corresponds to

$$\begin{equation} [F] = \lim_{p \in M^+ \to \Sigma} F(p) - \lim_{p \in M^- \to \Sigma} F(p)\tag{5} \end{equation}$$

The proof for this seems rather elusive, but according to Clarke and Dray, this stems from the fact that for $v$ some vector field such that $g(v, n) = 0$, with $n$ some extension of the normal form (I'm guessing via the normal bundle of the surface), we have

$$\begin{equation} v^c[g_{ab,c}] = v^c [g_{ab}]_{,c} = 0\tag{6} \end{equation}$$

which then implies that $[g_{ab,c}] = \gamma_{ab} n_c$. I'm not quite sure how to show this. Expanding everything, I get

$$\begin{equation} (\lim_{p \in M^+ \to \Sigma} \theta v^c g^+_{ab,c} - \lim_{p \in M^- \to \Sigma} (1 - \theta) v^c g^-_{ab,c})\tag{7} \end{equation}$$

given coordinates with tangent vectors $(n, \partial_\alpha)$, we can decompose this as

$$\begin{equation} v^c g^\pm_{ab,c} = v^\alpha g^\pm_{ab,\alpha}\tag{8} \end{equation}$$

since $v$ has no $n$ component. How to show that this quantity is then continuous upon crossing the boundary? Do I need to define the first fundamental form for every hypersurface $\Sigma_\varepsilon$ along the normal bundle of coordinate $\varepsilon$ and show that this is continuous?