Showing that the composition of maps of constant rank does not have to be of constant rank

Question

This is an exercise from p.79 of John Lee's Introduction to Smooth Manifolds. I am trying to search the smooth maps of the Euclidean space but cannot find a counterexample... Could anyone please suggest me one?

Answer 1

The map$f:\mathbb R\to \mathbb R^2:t \mapsto (t,t^2)\;$ has rank one everywhere.
The map $g:\mathbb R^2 \to \mathbb R:(x,y)\mapsto y\;$ has rank one everywhere.
Nevertheless the map $g\circ f:\mathbb R \to \mathbb R:t\mapsto t^2$ rank $1$ at $t\neq 0$ but rank $0$ at $t=0$ .