Let $$\mathbf{I}$$ be a continuous vector valued function in a volume $$V'$$ having boundary $$S'$$

Let $$\mathbf{r}$$ be the distance vector from points in $$V'$$ to a point $$P$$

Then the potential at point $$P$$ inside $$V'$$ due to a dipole distribution is:

$$\Phi = \int_{V'} \dfrac{\mathbf{I}.\mathbf{\hat{r}}}{r^2} dV' \tag1$$

Using spherical coordinate system, it can be seen that there is no integrand discontinuity.

$$\Phi = \int_{V'} \dfrac{\mathbf{I}.\mathbf{\hat{r}}}{r^2}\ r^2 \sin \theta'\ d\theta'\ d\phi\ dr'$$

Similarly, the field at point $$P$$ inside $$V'$$ due to a dipole distribution is:

$$\mathbf{E}=-\nabla \Phi=2 \int_V \dfrac{\mathbf{I}.\mathbf{\hat{r}}}{r^3}\ (\mathbf{\hat{r}})\ dV'$$

Using spherical coordinate system,

$$\mathbf{E}=-\nabla \Phi=2 \int_V \dfrac{\mathbf{I}.\mathbf{\hat{r}}}{r^3}\ (\mathbf{\hat{r}})\ r^2 \sin \theta'\ d\theta'\ d\phi\ dr'$$

the integrand discontinuity remains. So we have:

Statement I: $$\mathbf{E}$$ blows up at points inside $$V'$$

If $$\delta$$ is a small volume around point $$P$$, then applying vector identity $$\nabla.(a \mathbf{A})=\mathbf{A}.(\nabla a)+a (\nabla.\mathbf{A})$$ and the divergence theorem to equation (1):

\begin{align} \Phi &=-\int_{V'} \dfrac{\nabla.\mathbf{I}}{r} dV' + \lim \limits_{\delta \to 0} \int_{V'-\delta} \nabla . \left( \dfrac{\mathbf{I}}{r} \right) dV'\\ &=-\int_{V'} \dfrac{\nabla.\mathbf{I}}{r} dV' + \oint_{S'} \dfrac{\mathbf{I}.\mathbf{\hat{n}}}{r} dS' + \lim \limits_{\delta \to 0} \oint_{\partial \delta} \dfrac{\mathbf{I}.\mathbf{\hat{n}}}{r} dS'\\ &=-\int_{V'} \dfrac{\nabla.\mathbf{I}}{r} dV' + \oint_{S'} \dfrac{\mathbf{I}.\mathbf{\hat{n}}}{r} dS' \end{align}

Therefore:

\begin{align} \mathbf{E}&=-\nabla \Phi\\ &=-\int_{V'} \dfrac{\nabla.\mathbf{I}}{r^2} \mathbf{\hat{r}}\ dV' + \oint_{S'} \dfrac{\mathbf{I}.\mathbf{\hat{n}}}{r^2} \mathbf{\hat{r}}\ dS' \end{align}

Using spherical coordinate system, we can show that there is no integrand discontinuity in the first term.

Since $$P$$ is a point inside $$V'$$, the second term also has no discontinuity. So we have:

Statement II: $$\mathbf{E}$$ does not blow up at points inside $$V'$$

Statetements I and II are derived from equation (1). Yet there is a contradiction. Why?