Glaeser's continuity theorem

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 $f\ :\ U\rightarrow \mathbb {R} ^{+}$ be a function of class $C^{2}$ in an open set U contained in $\mathbb {R} ^{n}$ , 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.

References

^ G. Glaeser, "Racine carrée d'une fonction différentiable", Annales de l'Institut Fourier 13, no 2 (1963), 203–210 : article
^ J. Dieudonné, "Sur un théorème de Glaeser", J. Analyse math. 23 (1970), 85–88 : Résumé Zbl, article p.85, article p.86, article p.87 (the p. 88, not shown on the free preview contains the reference to Glaeser)

[1] G. Glaeser, "Racine carrée d'une fonction différentiable", Annales de l'Institut Fourier 13, no 2 (1963), 203–210 : article

[2] J. Dieudonné, "Sur un théorème de Glaeser", J. Analyse math. 23 (1970), 85–88 : Résumé Zbl, article p.85, article p.86, article p.87 (the p. 88, not shown on the free preview contains the reference to Glaeser)

[1]

[2]