$[0,1]$ is a $1$ manifold with boundary.
Observe that, since every subspace of a second countable Hausdorff space is second countable and Hausdorff, $[0,1]$ is second countable and Hausdorff, hence I must construct an appropriate chart for $[0,1]$.
Let $p\in [0,1]$. Let the chart be $([0,1], \psi(x))$ where $\psi: [0,1] \rightarrow \mathbb{R}$, $\psi(x)=2x$ is homeomorphism.
Clearly it is bijective and continuous, and it's inverse, $f: \mathbb{R} \rightarrow [0,1]$, $f(x)=\frac{x}{2}$ is also clearly continuous and must be bijective.
I have realized that my solution is incorrect, $\psi$ is not surjective. What homeomorphism would be more suitable?