Jtdylktuy

I am working in a problem of fluid dynamics (diffusion flames, exactly) and I am struggling to prove the continuity of a variable. The function $Z:R^nto R$ is continuous, but not necessarily smooth. Being $H(Z-Z_0)$ the Heaviside function, the equation for $Z$ is
$$
H(Z-Z_0) nabla cdot (mathbf{u} Z) = nabla^2 Z,
$$
with $nabla cdot mathbf{u}=sum_i delta(mathbf{r}-mathbf{r}_i)$, it is, $mathbf{u}$ is divergence-free almost everywhere, except for points inside the surface $Z=Z_0$ in which there are sources of $mathbf{u}$.

What can be said about the continuity of $nabla Z$ across the surface defined by $Z=Z_0$?

(If it's relevant, the boundary condition to $Z$ is piecewise continuous).

That is my attempt, assuming that $H(Z-Z_0)nabla Z = nabla [H(Z-Z_0) Z]$: in that case, the equation can be written as
$$
nabla cdot (mathbf{u} H(Z-Z_0) (Z-Z_1)) = nabla^2 Z,
$$
in which $Z_1$ is an arbitrary constant and, supposedly, does not afect the differential operator. Integrating the equation in the volume enclosed by $Z=C$,
$$
oint_{Z=C} H(Z-Z_0) (Z-Z_1) mathbf{u}cdot mathbf{n} dS = oint_{Z=C} nabla Z cdot mathbf{n} dS.
$$
Using $C=Z_0pm epsilon$ and taking the limit $epsilon to 0$ leads to
$$
oint_{Z=Z_0^+} (Z_0-Z_1) mathbf{u} cdot mathbf{n} dS = oint_{Z=Z_0^+} nabla Z cdot mathbf{n} dS,
$$
$$
0 = oint_{Z=Z_0^-} nabla Z cdot mathbf{n} dS.
$$
If $Z_1=Z_0$, $nabla Z$ is continuous across the surface $Z=Z_0$; otherwise, it's discontinuous. However, if $Z_1$ could be chosen arbitrarily, it would make no difference to the behaviour of $nabla Z$; if $Z_1=Z_0$ cannot be chosen, it's because $nabla(Z-Z_0) neq nabla Z$ and $nabla Z$ is not continuous at $Z=Z_0$.