Gradient of scalar function discontinuous on boundary

[mfurqan]

suppose g(r) is a scalar function which is constant inside the volume 'v' but discontinuous at the boundaries of 'v'. The magnitude of discontinuity is given by constant 'M' then can we write the following expression
$\int\nabla$g(r)dv=M$\int\hat{n}\delta$(r-rs)dv=M$\hat{n}\int$d$\delta$v

where $\delta$v is the boundary of volume 'v'
rs$\in\delta$v
$\hat{n}$ is the outward normal

 

[RedX]

I think your first integral is zero, even though there is a discontinuity at the boundary. The only possible way that your first integral could be non-zero is if the discontinuity jumped to infinity.

As for your second integral equaling your third integral, that's correct.