I start to think of this question when I attempt Ex 1.3.9 on Guillemin and Pollack's Differential Topology GP 1.3.9(b) Every manifold is locally expressible as a graph..
I am under the impression that this is true. For the proof of the following question, I need the fact that both the projection function and its inverse are smooth. Is this true?
Let $x_1, \dots, x_N$ be the standard coordinate functions on $\mathbb{R}^N$, and let $X$ be a $k$-dimensional submanifold of $\mathbb{R}^N$. Prove that every point $x \in X$ has a neighborhood on which the restrictions of some $k$-coordinate functions $x_{i_1}, \dots, x_{i_k}$ form a local coordinate system.