Let $(M,g)$ be a Riemannian manifold, and let $N\subset M$ be an embedded sphere that is everywhere smooth except for a single point at which the embedding will only be $C^0$.

How much regularity can I obtain for the square of the distance function $$ F(x) := \inf\{d^2(x,y)| y\in N\}? $$

I would be surprised if this extra information had some effect on the answer, but you never know: The embedding is in high codimension; $M$ is dimension $2n+1$ and $N$ is dimension $n-1$.

Thank you very much for every comment, help or reference.

Best wishes Klaus