Glaeser's continuity theorem explained

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} . It was introduced in 1963 by Georges Glaeser,^[1] and was later simplified by Jean Dieudonné.^[2]

The theorem states: Let

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

Notes and References

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