Consider a locally connected $X\subset \mathbb R^n$. Given a point $p\in X$, consider the following condition:
- For any $a,b\in X$, if there's a path $a\to p\to b$ in $X$, there is also a smooth path $a\to p \to b$ in $X$ with nonzero derivative at $p$ (at least once).
I think this condition may express that $X\subset\mathbb R^n$ is smooth at $p$.
Every point $p$ of Euclidean space satisfies this property since e.g the circle going through points $a,b,p$ furnishes a smooth path $a\to p\to b$. Consequently I think every point of an embedded manifold $X\subset\mathbb R^n$ satisfies this condition.
Question. Does this characterize manifolds embedded in Euclidean space? That is, if $X\subset \mathbb R^n$ is a locally connected subset whose every point satisfies the above condition, does it follow $X$ is an embedded manifold?
Added. Eric Wofsey's answer helped me realize I would like to assume $X$ is additionally locally Euclidean. His comment provides a counterexample to this case, namely the wedge in $\mathbb R^3$ formed by folding a rectangle along a line. This comment outlines the problem with the condition of my question. I have asked a follow-up question here.