We consider the set of points x ∈ Rn+1 satisfying Hx = 0, where H: Rn+1 → Rn is a C1 function and 0 is a regular value. This set, H-10, is a C1 one-dimensional manifold, and each component can be described by a curve xθ. In this note a theorem is proved which is directly related to and motivated by a result due to Eaves and Scarf on piecewise linear functions. This theorem relates the signs of the derivatives xtθ to the signs of the determinants of submatrices of the Jacobian matrix H'. Applications to solving nonlinear equations are given.
['C. B. Garcia', 'Floyd J. Gould']
A Theorem on Homotopy Paths