Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 1 / 24H / i / Solution

A point $x\in X$ is a critical point of a smooth map if $d{f}_x:T_{x}X\to T_{f(x)}Y$ is not surjective. A critical value is the image of at least one critical point. A regular value is a point $y\in Y$ for which every point of $f^{-1}(y)$ has surjective differential. A point outside the image is therefore a regular value vacuously. These definitions use rank relative to $\dim Y$, not merely whether a derivative vanishes in chosen coordinates.