This is a follow up question (which is unanswered in math.SE) for a previous one. Consider the following functions $F_{ij}:S\subset{\mathbb R}^3\to{\mathbb R}$, $$ F_{ij}(y) = \begin{cases} \frac{(y_i-x_i)(y_j-x_j)(y-x)\cdot n(y)}{|y-x|^3},&y\neq x; \\ 0,& y=x.\end{cases} \quad i,j = 1,2,3 $$ where $S$ is a surface which has a continuously varying normal vector, $x=(x_1,x_2,x_3)\in S$ is given, $y=(y_1,y_2,y_3)\in S$, $n(y)$ is the normal vector at point $y$ which is assumed to be smooth. Here $(y-x)\cdot n(y)$ is the dot product. Using the method in the answer to the previous question, I conclude that given $i,j$ $$ \lim_{y\to x}F_{ij}(y)=0 $$ which implies that $F_{ij}(y)$ is continuous at $y=x$.

Here is my **question**:

- Is $F_{ij}(y)$
smoothat $x$? If it is not, what would be the key properties to fail the smoothness?

An immediate idea is that I should test the smoothness of $F_{ij}$ by definition. The difficulty is that with a parameterization $y=y(\alpha,\beta)$, it is not trivial to find the high order partial derivatives for $F_{ij}(y(\alpha,\beta))$. I am not even able to determine whether $F_{ij}$ is $C^1(S)$ or not.