Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 3 / 23H / i / Solution

A point $x\in X$ is critical if $d{f}_x:T_{x}X\to T_{f(x)}Y$ is not surjective. A critical value is an image of a critical point; a regular value is a point $y\in Y$ for which every point of $f^{-1}(y)$ is regular, including vacuously values with empty preimage. Sard theorem says that for a smooth map of finite-dimensional manifolds, the critical values have measure zero in every coordinate chart of the target. In particular regular values are dense.