I am reading "Analysis on Manifolds" by James R. Munkres.
Lemma 23.3. Let $M$ be a manifold in $\mathbb{R}^n$, and let $\alpha:U\to V$ be a coordinate patch on $M$. If $U_0$ is a subset of $U$ that is open in $U$, then the restriction of $\alpha$ to $U_0$ is also a coordinate patch on $M$.
Note that this result would not hold if we had not required $\alpha^{-1}$ to be continuous. The map $\alpha$ of Example 3 satisfies all the other conditions for a coordinate patch, but the restricted map $\alpha|U_0$ is not a coordinate patch on $M$, because its image is not open in $M$.
The author says "if we had not required $\alpha^{-1}$ to be continuous", then Lemma 23.3 "would not hold".
But Lemma 23.3 requires that $\alpha^{-1}:V\to U$ is continuous because Lemma 23.3 requires $\alpha$ is a coordinate patch.
I cannot understand what the author wants to say.
Munkres's remark refers to the concept of coordinate patch introduced in the definition at the beginning of §23.
A coordinate patch $\alpha$ on $M$ is defined as a contiuous bijection $\alpha : U \to V$ between an open $U \subset \mathbb R^k$ and an open $V \subset M$ such that
$\phantom{xx} (1)\phantom{x}$ $\alpha$ is of class $C^r$.
$\phantom{xx} (2)\phantom{x}$ $\alpha^{-1} : V \to U$ is continuous.
$\phantom{xx} (3)\phantom{x}$ $D\alpha(\mathbf x)$ has rank $k$ for each $\mathbf x \in U$.
Condition $(2)$ means that $\alpha$ (which is a bijection!) is an open map.
One could alternative require that $\alpha$ is a homeomorphism satisfying $(1)$ and $(3)$.
In the definition on p.201 he generalizes this to introduce manifolds with boundary.
Concerning Lemma 23.3 Munkres writes
What he wants to say that if we had omitted $(2)$ in the definition of a coordinate patch, then we could not conclude that $\alpha(U_0)$ is open in $M$ which is a necessary requirement for the restriction of $α$ to $U_0$ being a coordinate patch on $M$.