Consider an oriented $C^1$ surface ${\cal S}$ whose closure is a $C^0$ surface with boundary whose boundary is a $C^1$ curve. If the normal to ${\cal S}$ is uniformly continuous, so that it has a continuous extension to the closure of ${\cal S}$, does this imply that the closure of ${\cal S}$ is a $C^1$ manifold with boundary?
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$\begingroup$ I suspect this is related to your other question: mathoverflow.net/questions/464331/… i.e. you are interested in 2-manifolds in $\mathbb R^3$. $\endgroup$– Ryan BudneyCommented Feb 18 at 17:54
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