Quote An introduction to manifolds:
In the pictures $f(t) = (\cos t,\sin 2t)$ and $g(t) = (\cos t,−\sin 2t)$
I'm completely confused here, he is proving that $\overline{f}$ is not continuous if $S$ has the induced topology from $g$. But this seems like cheating really, and $S$ is not even a manifold in the subspace topology of $\mathbb{R}^2$.
Is there an honest example, where $S$ is a manifold in the subspace topology and $f:N\to S$ is not smooth? I know that if in addition $S$ is a regular submanifold then it is always true.