a while ago I was studying about manifolds and this question gave me some troubles:
Let $f:U \to \Bbb{R}$ a $C^{1}$ class function, $U \subset \Bbb{R}^{n}$ an open. Prove the following:
I: The graph of $f$ is a differential and orientable manifold in $\Bbb{R}^{n+1}$;
II: The tangent space of $graph(f)$ is given by $T_{a}M = \{a\}\times \{(u,v) \in \Bbb{R}^{n} \times \Bbb{R}; v = \nabla f(a)u \}$;
III: Every embedded manifold can by written (localy) as a graph of a function.
So, any help you could give, I'll be extremely grateful. I think that the idea of the question is to prove III with I and II.