Skip to main content
SpringerLink
Log in

Implicit functions and diffeomorphisms without C 1

  • Chapter
Advances in Mathematical Economics

Part of the book series: Advances in Mathematical Economics ((MATHECON,volume 5))

Abstract

We prove implicit and inverse function theorems for non-C 1 functions, and characterize non-C 1 diffeomorphisms.

We are indebted to Professor Wayne H. Richter, University of Minnesota; Professor Hiilya Eraslan, University of Pennsylvania; and Nevzat Eren, University of Minnesota, for valuable comments on an earlier version.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Berge, C: Espaces Topologique. Dunod, Paris 1959

    Google Scholar 

  2. Berge, C: Topological Spaces. Macmillan, New York 1963

    Google Scholar 

  3. Bliss, G.A.: Fundamental existence theorems. In: Lectures on Mathematics, pp. 1–107 American Mathematical Society 1913 (The Princeton Colloquium. Lectures delivered September 15 to 17, 1909)

    Google Scholar 

  4. Bolza, O.: Vorlesungen über Variationsrechnung. Chelsea Publishing Company, New York (second edition, no date, author’s preface dated, 1909)

    Google Scholar 

  5. Briot, C.A.A., Bouquet, C: Théorie des fonctions doublement périodiques et, en particulier des fonctions elliptiques. Gauthiers-Villars, Paris 1875 (Deuxième Édition)

    Google Scholar 

  6. Brouwer, L.E.J.: Beweis der Invarianz des n-dimensionalen Gebiets. Mathematische Annalen 71, 305–313 (1912)

    Article  Google Scholar 

  7. Brouwer, L.E.J.: Über Abbildung von Mannigfaldigkeiten. Mathematische Annalen 71, 97–115 (1912)

    Article  Google Scholar 

  8. Carathéodory, C: Calculus of Variations and Partial Differential Equations of the First Order. Chelsea Publishing Company, New York, English edition, 1982 (Second (revised) edition originally published as Variationsrechnung und Partielle Differentialgleichungen erster Ordnung, B. G. Teubner, Berlin, 1935)

    Google Scholar 

  9. Cauchy, A.: Mémoir sur l’application du calcul infinitésimal à la détermination des fonctions implicites. In: Oeuvres Complètes d’Augustin Cauchy, volume 11, pp. 406–415. 1899. Edited by L’Académie des Sciences; C.R., T. 34, p. 265 (23 février 1852).

    Google Scholar 

  10. Cauchy, A.: Résumé d’un mémoir sur la mécanique céleste et sur un nouveau calcul appelé calcul des limites. In: Oeuvres Complètes d’Augustin Cauchy, volume 12 of II, pp. 48-112. Gauthier-Villars, Paris, 1916. Edited by L’Académie des Sciences. The part pertaining to series expansions was previously read at a meeting of the Académie de Turin, 11 October, 1831.

    Google Scholar 

  11. Conway, J.B.: Functions of One Complex Variable. Springer-Verlag, New York, second edition, 1978.

    Google Scholar 

  12. Debreu, G.: A social equilibrium existence theorem. In: Proceedings of the National Academy of Sciences, volume 38, pp. 886–893 1952

    Article  Google Scholar 

  13. Debreu, G.: Theory of Value. John Wiley and Sons, New York 1959

    Google Scholar 

  14. Dieudonné, J.: Foundations of Modern Analysis. Academic Press 1969 (Enlarged and Corrected Printing)

    Google Scholar 

  15. Dini, U.: Analisi Infinitesimale. Litografia Gorani, Pisa 1877–1878 (Lithographed version of manuscript lecture notes)

    Google Scholar 

  16. Evgrafov, M.A.: Analytic Functions. W.B. Saunders Company, Philadelphia 1966

    Google Scholar 

  17. Genocchi, A.: Calcolo Differenziale e Principii di Calcolo Integrale. Fratelli Bocca, Turin 1884 (Pubblicato con aggiunte dal Dr. Guiseppe Peano)

    Google Scholar 

  18. Genocchi, A.: Differentialrechnung und Grundzüge der Integralrechnung. Teubner, Leipzig 1899 (German translation by G. Bohlman and A. Schepp)

    Google Scholar 

  19. Goursat, É.: Sur la théorie des fonctions implicites. Bulletin de la Société Mathématique de France 31, 184–192 (1903)

    Google Scholar 

  20. Goursat, É.: A Course in Mathematical Analysis, volume I. Ginn and Company, Boston 1904. Translated by E. R. Hedrick.

    Google Scholar 

  21. Goursat, É.: A Course in Mathematical Analysis, volume II, Part 1. Ginn and Company, Boston 1916. Translated by E. R. Hedrick and Otto Dunkel.

    Google Scholar 

  22. Graves, L.M.: The Theory of Functions of Real Variables. McGraw-Hill, New York 1956 (second edition)

    Google Scholar 

  23. Halkin, H.: Implicit functions and optimization problems without continuous differentiability of the data. Siam Journal on Control 12, 229–236 (1974)

    Article  Google Scholar 

  24. Hildebrand, H., Graves, L.M.: Implicit functions and their differentials in general analysis. Transactions of the American Mathematical Society 29, 127–153 (1927)

    Article  Google Scholar 

  25. Hurwicz, L.:, Richter, M.K.: Implicit Functions and Diffeomorphisms without C 1. Discussion Paper No. 279. Department of Economics, University of Minnesota 1994

    Google Scholar 

  26. Hurwicz, L., Richter, M.K:. Optimization and Lagrange Multipliers: Non-C 1 Constraints and “Minimal” Constraint Qualifications. Discussion Paper No. 280. Department of Economics, University of Minnesota 1995 (see also this volume, pp.97-151)

    Google Scholar 

  27. Jordan, C: Cours d’analyse de l’École Polytechnique. (Tome Premier, Calcul differential, deuxième edition.) Gauthier-Villars et Fils, Paris erne edition 1893

    Google Scholar 

  28. Leach, N.B.: A note on inverse function theorems. Proceedings of the American Mathematical Society 12, 694–697 (1961)

    Article  Google Scholar 

  29. Lindelöf, E.: Demonstration élémentaire de l’existence des fonctions implicites. Bulletin des Sciences Mathématique 23, 68–75 (1899) (Deuxième Série)

    Google Scholar 

  30. Lindelöf, E.: Le Calcul des Résidus et ses Applications à la Théorie des Fonctions. Chelsea Publishing Company, New York 1947 (Preface dated November 13, 1904)

    Google Scholar 

  31. Markushevich, A.I.: Theory of Functions of a Complex Variable, volume II. Prentice Hall, Englewood Cliffs, NJ 1965 (English edition) Revised translated and edited by R. A. Silverman.

    Google Scholar 

  32. Nijenhuis, A.: Strong derivatives and inverse mappings. American Mathematical Monthly 81, 969–980 (1974)

    Article  Google Scholar 

  33. Osgood, W.F.: Analytische Funktionen komplexer Grössen. In: Encyklopädie der Mathematischen Wissenschaften, volume Zweiter Teil. Analysis, pp. 1–114. Teubner, Leipzig 1901-1921 (Originally published in “Heft 1,” whose date was 27.XII.1901)

    Google Scholar 

  34. Osgood, W.F: Lehrbuch der Funktionentheorie, volume I. Teubner, Leipzig 1907

    Google Scholar 

  35. Osgood, W.F.: Lehrbuch der Funktionentheorie, volume II. 1. Teubner, Leipzig 1929 (second edition)

    Google Scholar 

  36. Radulescu, S., Radulescu, M.: Local inversion theorems without assuming continuous differentiability. Journal of Mathematical Analysis and Applications 138, 581–590(1989)

    Article  Google Scholar 

  37. Samuelson, P.A.: Foundations of Economic Analysis. Harvard University Press 1947

    Google Scholar 

  38. Weierstrass, K.: Abhandlungen aus der Functionenlehre. Julius Springer, Berlin 1886

    Google Scholar 

  39. Young, W.H.: On implicit functions and their differentials. In: Proceedings of the London Mathematical Society, volume 7, pp. 397–421 1909 (Second Series)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Economics, University of Minnesota, 271 19th Avenue South, Minneapolis, MN, 55455, USA

    Leonid Hurwicz & Marcel K. Richter

Authors
  1. Leonid Hurwicz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marcel K. Richter
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-0041, Japan

    Shigeo Kusuoka (Professor) (Professor)

  2. Department of Economics, Keio University, 2-15-45 Mita, Minato-ku, Tokyo, 108-8345, Japan

    Toru Maruyama (Professor) (Professor)

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Tokyo

About this chapter

Cite this chapter

Hurwicz, L., Richter, M.K. (2003). Implicit functions and diffeomorphisms without C 1 . In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics. Advances in Mathematical Economics, vol 5. Springer, Tokyo. https://doi.org/10.1007/978-4-431-53979-7_4

Download citation

  • DOI: https://doi.org/10.1007/978-4-431-53979-7_4

  • Received:

  • Revised:

  • Publisher Name: Springer, Tokyo

  • Print ISBN: 978-4-431-00003-7

  • Online ISBN: 978-4-431-53979-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation