Ukrainian Mathematical Journal

, Volume 59, Issue 2, pp 243–292 | Cite as

Some results of the local theory of smooth functions

  • A. M. Samoilenko
Article

Abstract

We present results of the investigation of the local behavior of smooth functions in neighborhoods of their regular and critical points and prove theorems on the mean values of the functions considered similar to the Lagrange finite-increments theorem. We also study the symmetry of the derivative of an analytic function in the neighborhood of its multiple zero, prove new statements of the Weierstrass preparation theorem related to the critical point of a smooth function with finite smoothness, determine a nongradient vector field of a function in the neighborhood of its critical point, and consider one critical case of stability of an equilibrium position of a nonlinear system.

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Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • A. M. Samoilenko
    • 1
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKiev

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