Regular Values

Many interesting and important properties of C r functions of manifolds into manifolds, as in the case of functions of euclidean spaces into euclidean spaces, do not hold in full generality, but only under appropriate restrictions. For example, most, but not all, the level sets of a C r function exhibit a manifold structure (if they are not empty). The concept of regular value provides in this case the appropriate dividing line, as stated by the regular value theorem. This is the object of the first section, which contains the definition of regular and critical values, followed by a discussion of some their properties, the theorem’s statement and proof, and an interpretation of the result in terms of nonlinear systems of equations. In the same spirit, the concept of proper function is introduced and discussed in the second section. The assumption of properness allows us to sharpen our understanding the structure of solution sets to nonlinear systems of equations. Even when such an assumption is not granted, some of the conclusions achieved in the properness case can be seen to hold; this is the object of the third section. The final section contains a few results based on regularity and properness which are heavily used in the second part of the book.