Let $M$ be a non-empty subset of $\mathbb R^n$, $n \geq 2$.

Recall that a vector $v$ is tangent to $M$ at the point $m \in M$ if it exists a differentiable curve $\gamma : I \to M$ such that $\gamma(0) = m$ and $\gamma'(0) = v$, where $I \subset \mathbb R$ is an interval that contains a neighborhood of $t=0$.

Suppose that it exists an integer $k \geq 1$ such that, for every $m \in M$, the set of vectors that are tangent to $M$ at $m$ is a linear space of dimension $k$ (the same $k$ for every $m$).

Is it true that $M$ is a differential manifold of dimension $k$?

domean $M$ to be a general subset of $\mathbb{R}^n$, not necessarily open. I read too quickly and gave an unhelpful link. :) $\endgroup$