How do I prove that a subset of a manifold is not a submanifold?

Solution 1:

If $S\subset M$ is a subset of the manifold $M$, then it has a tangent cone $C(s)$ at every $s\in S$, which consists of the tangent vectors at $0$ of smooth curves $\gamma:I\to S$, where $ I\subset \mathbb R$ is an open interval containing $0 $ .
A necessary condition for $S$ to be a submanifold of dimension $k$ is that $C(s)$ be a vector space of dimension $k$ at every $s\in S$.

An example
If $M=\mathbb R^2$ and $S$ is the graph of the function $|x|$, then at $O=(0,0)$ the cone $C(O)$ is the zero vector space and at every other point of $s\in S$ the tangent cone $C(s)$ is $1$-dimensional.
Thus $S$ is not a submanifold of $M$.