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

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