Divergence theorem with non smooth functions

Question

How do you extend the classical multivariable integration by parts formula for functions with countably many discontinuities?

Is the result still true? Can I use an approximating sequence?

Answer 1

It is true. Here's another way of seeing it. By the divergence theorem, $$ \int_\Omega div(f\phi)\,\mathrm{d}x=\int_{\partial \Omega}f\phi\cdot \nu\,\mathrm{d}S, $$ where $\nu$ is the outward unit normal. Since $\phi$ has compact support, the right hand side equals zero. Then component wise, this means $$ \int_\Omega \partial_i (f\phi)\,\mathrm{d}x=\int_\Omega \partial_i(f)\phi\,\mathrm{d}x+\int_\Omega f\partial_i(\phi)\,\mathrm{d}x=0. $$