Global Implications of the Implicit Function Theorem

Abstract

Sensitivity analysis has long been an important technique in science: How sensitive is the international price equilibrium to the availability of oil? How does the strain on a bridge depend upon the weight carried? Or, how does the incidence of an infectious desease depend upon the fraction of the population that is checked daily? These questions can be answered by determining how solutions of systems of equations depend on a parameter. Does the solution continue as the parameter is varied? If it does, how far through the parameter range does it extend? Formally, let F: Rⁿ⁺¹ → Rⁿ be a C¹ map. (In practice, f is usually defined on a subset of Rⁿ⁺¹.) We denote by (x,α) a point in Rⁿ⁺¹ = Rⁿ × R, where x ∈ Rⁿ and α ∈ R (a scalar parameter), and let

$${\rm{C}}\,{\rm{ = }}\,{\rm{\{ (X,}}\,{\rm{\alpha )}}\, \in \,{{\rm{R}}^{{\rm{r}}\,{\rm{ + 1}}}}:\,{\rm{f}}\,{\rm{(X,}}\,{\rm{\alpha )}}\, = \,0\} $$

be the set of zeroes of f. When the implicit function theorem is applied at a point (x̄,ᾱ) ∈ C, one learns about the local structure of C near (x̄,ᾱ), and thus how small changes in the parameter α affect the set of zeroes. The conclusion of this theorem, however, says nothing about the set of zeroes globally (i.e., over the entire parameter range). Concentrating on the (connected) component of C that contains (x̄,ᾱ), we show that the hypotheses of the implicit function theorem imply certain global facts about this component.