Lee in Introduction to Topological Manifolds defines regular coordinate balls as follows:
"Let $M$ be a $n$-manifold. We say that a coordinate ball $B\subset M$ is a regular coordinate ball if there is a neighbourhood $B'$ of $\overline{B}$ and a homeomorphism $\varphi:B'\to B_{r'}(x)\subset \mathbb{R}^n$ that takes $B$ to $B_r(x)$ and $\overline{B}$ to $\overline{B_r(x)}$ for some $r'>r>0$ and $x\in\mathbb{R}^n$."
I was wondering if this means that $B\cong B_r(x)$ and $\overline{B}\cong\overline{B_r(x)}$? I know that restricting $\varphi$ to $B$, I get a continuous injective map $B\to B_{r'}(x)$, which gives a continuous bijection $B\to B_r(x)$. Here I used $f:X\to Y$ continuous, $A\subset X$ implies $f|_{A}:A\to f(A)$ is continuous. Similarly, I can show there is a continuous bijection $\overline{B}\to \overline{B_r(x)}$ upon restricting $\varphi$ to $\overline{B}$.
I am having trouble showing that the inverse map is also continuous. Any suggestions will be greatly appreciated!!