A note on the interrelation of subsets of independent variables of a continuous function with continuous first derivatives

Let X be a set of n variables xi, # 2 , , x n and y a continuous function of these variables, 3>=/(xi, #2, , x n ), also written as f(X) with continuous partial derivatives of the first, second, and third orders. Throughout the following discussion all the first partial derivatives of f(xi, X2, - , x n ) are assumed to be different from zero.