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Glaeser's continuity theorem

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Characterizes the continuity of the derivative of the square roots of C2 functions

In mathematical analysis, Glaeser's continuity theorem is a characterization of the continuity of the derivative of the square roots of functions of class C 2 {\displaystyle C^{2}} . It was introduced in 1963 by Georges Glaeser, and was later simplified by Jean Dieudonné.

The theorem states: Let f   :   U → R 0 + {\displaystyle f\ :\ U\rightarrow \mathbb {R} _{0}^{+}} be a function of class C 2 {\displaystyle C^{2}} in an open set U contained in R n {\displaystyle \mathbb {R} ^{n}} , then f {\displaystyle {\sqrt {f}}} is of class C 1 {\displaystyle C^{1}} in U if and only if its partial derivatives of first and second order vanish in the zeros of f.

References

  1. Glaeser, Georges (1963). "Racine carrée d'une fonction différentiable". Annales de l'Institut Fourier. 13 (2): 203–210. doi:10.5802/aif.146.
  2. Dieudonné, Jean (1970). "Sur un théorème de Glaeser". Journal d'Analyse Mathématique. 23: 85–88. doi:10.1007/BF02795491. Zbl 0208.07503.
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