Does every manifold have an atlas consisting of graph coordinates?

Question

A pointer to a reference will suffice for an answer.

1. Given an $n$-dimensional smooth manifold $M$, do we have an atlas $\{(U, \phi)\}$ for $M$ such that each $\phi$ has the form: $$\phi(p)=(x^1, \dots, x^n)=(y^1, \dots, y^n, f(y_1, \dots, y^n)) $$ for a smooth function $f: \mathbb{R}^n \to \mathbb{R}$? I.e. can the manifold be "divided into pieces", such that each "piece" is diffeomorphic to the graph of a smooth function?

2. Does the analogous statement hold for topological manifolds? (Replacing "smooth manifold" by "topological manifold" and "smooth function" by "continuous function".)

3. If a second countable, Hausdorff space is locally expressible as the graph of a continuous function, then is it a topological manifold?

(Yes, because the identification $(x^1, \dots, x^n) \leftrightarrow (y^1, \dots, y^n, f(y^1, \dots, y^n))$ is a homeomorphism?)

Note: this question is a duplicate of this unanswered question. For smooth manifolds, the answer obviously involves the implicit function theorem.

However, I am not only considering manifolds which are already explicitly embedded or immersed in $\mathbb{R}^n$. (I.e. I am asking about intrinsic geometry of manifolds not extrinsic.) To the best of my understanding, an embedding could be used to construct such an atlas, but a general immersion cannot. If this understanding is correct, then the answer to 1. in the affirmative follows from Whitney's theorem.)

Answer 1

As was pointed out in the comments, this question only makes sense with reference to an explicit choice of embedding of the manifold in Euclidean space. For second countable manifolds, such an embedding always exists by the Whitney embedding theorem.

Anyway, referring to manifolds explicitly embedded in Euclidean space, it is true that they are all locally the graphs of smooth functions, as a corollary of the implicit function theorem.

The best reference for this fact I could find is Section 4.3 of Krantz, Parks's Implicit Function Theorem. See also this answer on MathOverflow.

This is also addressed on p. 196, Section III.4, of Edwards's Advanced Calculus of Several Variables:

Recall that a $k$-dimensional manifold in $\mathbb{R}^n$ is a set $M$ that looks locally like the graph of a mapping from $\mathbb{R}^k$ to $\mathbb{R}^{n-k}$. That is, every point of $M$ lies in an open subset $V$ of $\mathbb{R}^n$ such that $P = V \cap M$ is a $k$-dimensional patch. Recall that this means there exists a permutation $x_{i_1}, \dots, x_{i_n}$ of $x_1, \dots, x_n$, and a differentiable mapping $h: U \to \mathbb{R}^{n-k}$ defined on an open set $U \subset \mathbb{R}^k$, such that $$P = \{ x \in \mathbb{R}^n: (x_{i_1}, \dots, x_{i_k}) \in U \quad \text{and} \quad (x_{i_{k+1}}, \dots, x_{i_n}) = h(x_{i_1}, \dots, x_{i_k}) \}. $$