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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?

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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 $t=0$ .

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    $\begingroup$ I finally managed to find one myself and actually it is similar to yours. Thank you anyway. $\endgroup$
    – Keith
    Feb 2, 2018 at 9:24
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    $\begingroup$ Congratulations Keith: good minds think alike :-) $\endgroup$ Feb 2, 2018 at 13:16

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