We have a smooth embedded submanifold $M\subseteq\mathbb{R}^n$ and we want to show that there is a $c\in \mathbb{R}$ such that the plane $x_1=c$ intersects $M$ transversally. Show that the set of $c$ is not necessarily open in $\mathbb{R}$.
I have beat my head against this problem, but while I know the definition of intersecting transversally (submanifolds of $Y,$ $M$ and $N$ intersect transversally if $\forall x\in M\cap N,$ $T_x M+T_x N=T_x Y$) I really struggle to work with this definition.