Counterexample Immersed Submanifold.

Let $M$ be a manifold.

Let $N$ be a subset of $M$ and suppose that $N$ is endowed with a manifold structure for which the inclusion $i: N \rightarrow M$ is a smooth map. Is $N$ necessarily an immersed submanifold of $M$?

My definition of an immersed submanifold is the following: an immersed submanifold of $M$ is a subset $H \subset M$ with a topology and a smooth manifold structure such that the inclusion $i: H \rightarrow M$ is an immersion.

Looking at the definition of an immersed submanifold, it looks clear to me that the statement is not true since the derivative of the inclusion at every point has to be injective. However, I can't come up with a counterexample. Furthermore, I have trouble to find examples of inclusions whose derivative at a point is not injective. Someone who can given me a clear example? Thanks in advance!

Let $M = \mathbb R^2$ and $N = \{ (x, 0)\in M|\ x\in \mathbb R\}$ be the x axis. Then $\{(N, \varphi)\}$ with $\varphi : N\to \mathbb R$, $\varphi (x, 0) = x^{1/3}$ is a smooth atlas on $N$. Let $\iota : N \to M$ be the inclusion, then under the local chart $(N, \varphi)$ and $(\mathbb R^2, \mathrm{id})$ on $M = \mathbb R^2$,

$$ \mathrm{id}_M \circ \iota \circ\varphi^{-1} (t) = (t^3, 0).$$ Thus $\iota$ is smooth but not immersed.