The measure of the critical values of differentiable maps

of a region R of euclidean m-space into part of euclidean w-space. Suppose that each f unction ƒ' 0' = 1, • • • , n) is of class C in R (q^l). A critical point of the map (1.1) is a point in R at which the matrix of first derivatives 2)? = ||/*|| (i = ly • • • , m;j = l, • • • , n) is of less than maximum rank. The rank of a critical point # is the rank of 5DÎ at x. A critical value is the image under (1.1) of a critical point. If » = 1, these definitions are the usual definitions of critical point and value of a continuously differentiable function. We prove the following result: If m^n, the set of critical values of the map (1.1) is of m-dimensional measure zero without further hypothesis on q; if m>n, the set of critical values of the map (1.1) is of n-dimensional measure zero providing that q^m — n + 1. Using an example due to Hassler Whitney we show that the hypothesis on q cannot be weakened. We prove also that the critical values of (1.1) corresponding to critical points of rank zero constitute a set of (m/q)dimensional measure zero. The idea of considering the measure of the set of critical values of one function or of several functions is due to Marston Morse. The first result stated above reduces, if » = 1, to the known theorem : The critical values of a function of m variables of class C constitute a set of linear measure zero. A. P. Morse has given a proof of this theorem for all m. In the present paper we make use of one of A. P. Morse's results.