TITLE: Is the gradient of the self-intersections of a curve zero?
QUESTION [5 upvotes]: Suppose a curve with self-intersections can be described by $\phi(x,y)=0$. Suppose the intersections are $T_i$, $i=1,2,...$ and the gradient $\nabla \phi$ at those intersections are well defined. Then is it true that $\nabla\phi(T_i)=0$ for all $i$? In other words, are the gradients at those intersections all zero?

REPLY [4 votes]: A different argument
(For the case where two branches of the curve have distinct tangents at the intersection point)
Since $\phi$'s values do not change along the curve, we know that if $\phi$ is continuously differentiable then any of its directional derivatives along the curve is $0$. 
At an intersection point, we would have a null directional derivative along two linearly independent directions. Having projection $0$ along two linearly independent vectors, we know the gradient is $0$.