Consider a spherical coordinate system with origin at point $O$. Consider the following volume integral over $V$:
$\displaystyle \int_V \nabla \cdot \left( \dfrac{\mathbf{A}}{r} \right) dV$
where $\mathbf{A}$ is a continuously differentiable vector field.
Let $O \in V$.
$\dfrac{\mathbf{A}}{r}$ is differentiable everywhere except at origin $O$.
Therefore the volume integral has undefined output at one point in its domain.
Will this mere fact prevent us from computing the Riemann integral (by finding the antiderivative and applying the limits)?
I think I have read somewhere that if we redefine the undefined output at one point in such a way that the function becomes continuous, we can proceed to do the Riemann integral (by finding the antiderivative and applying the limits). Am I correct?