If $α:U→V$ is a coordinate patch on a manifold without boundary, $M$, about the point $p$, one of the conditions on $α$ is that $Dα(x)$ have rank 1 for each $x \in U$. According to Munkres in Analysis on Manifolds, this condition “rules out the possibility that $M$ could have ‘cusps’ and ‘corners.’”
But let $M = \{ (x,y) \mid y = |x|\}$. For any set $V$ open in $M$ containing a point $p \in M$, Let $α: U→V$ be defined by $α(t) = (t, |t|)$, where $U$ is an appropriately defined open set in $\mathbb{R}$.
$M$ has a corner at the origin. And it appears that the coordinate patch $α$ satisfies the condition that $Dα(x)$ have rank 1 $\forall x \in U$ (as well as the other conditions for a coordinate patch).
Is this therefore an example of a simple 1-manifold in $\mathbb{R}^2$ with a “corner”? Or am I missing something?