Abstract
The idea of solving an equation f(p,x) = 0 for x as a function of p, say x = s(p), plays a huge role in classical analysis and its applications. The function obtained in this way is said to be defined implicitly by the equation. The closely related idea of solving an equation f(x) = y for x as a function of y concerns the inversion of f. The circumstances in which an implicit function or an inverse function exists and has properties like differentiability have long been studied. Still, there are features which are not widely appreciated and variants which are essential to seeing how the subject might be extended beyond solving only equations. For one thing, properties other than differentiability, such as Lipschitz continuity, can come in. But fundamental expansions in concept, away from thinking just about functions, can serve in interesting ways as well.
As a starter, consider for real variables x and y the extent to which the equation x 2 = y can be solved for x as a function of y. This concerns the inversion of the function f(x) = x 2 in Figure 1.1 below, as depicted through the reflection that interchanges the x and y axes. The reflection of the graph is not the graph of a function, but some parts of it may have that character. For instance, a function is obtained from a neighborhood of the point B, but not from one of the point A, no matter how small.
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Notes
- 1.
Ulisse Dini (1845–1918). Many thanks to Danielle Ritelli from the University of Bologna for a copy of Dini’s manuscript.
- 2.
These two proofs are not really different, if we take into account that the contraction mapping principle is proved by using a somewhat similar iterative procedure, see Section 5E.
- 3.
Isaac Newton (1643–1727). In 1669 Newton wrote his paper De Analysi per Equationes Numero Terminorum Infinitas, where, among other things, he describes an iterative procedure for approximating real roots of a polynomial equation of third degree. In 1690 Joseph Raphson proposed a similar iterative procedure for solving more general polynomial equations and attributed it to Newton. It was Thomas Simpson who in 1740 stated the method in today’s form (using Newton’s fluxions) for an equation not necessarily polynomial, without making connections to the works of Newton and Raphson; he also noted that the method can be used for solving optimization problems by setting the gradient to zero.
- 4.
A set C such that P C is single-valued is called a Chebyshev set. A nonempty, closed, convex set is always a Chebyshev set, and in IR n the converse is also true; for proofs of this fact see Borwein and Lewis [2006] and Deutsch [2001]. The question of whether a Chebyshev set in an arbitrary infinite-dimensional Hilbert space must be convex is still open.
- 5.
These two examples are from Nijenhuis [1974], where the introduction of strict differentiability is attributed to Leach [1961]. By the way, Nijenhuis dedicated his paper to Carl Allendoerfer “fornot taking the implicit function theorem for granted.” In the book we follow this advice.
- 6.
Theorem 1F.3 can of course be proved directly, without resorting to Brouwer’s theorem 1F.1.
- 7.
The left inverse and the right inverse are particular cases of the Moore-Penrose pseudo-inverseA + of a matrixA. For more on this, including the singular-value decomposition, see Golub and Van Loan [1996].
- 8.
Edouard Jean-Baptiste Goursat (1858–1936). Goursat paper from 1903 is available at http://www.numdam.org/.
- 9.
The name “Banach spaces” for normed linear spaces that are complete was coined by Fréchet, according to Hildebrand and Graves [1927]; we deal with Banach spaces in Chapter 5.
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Dontchev, A.L., Rockafellar, R.T. (2009). Functions Defined Implicitly by Equations. In: Implicit Functions and Solution Mappings. Springer Monographs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-87821-8_1
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