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, and was later simplified by Jean Dieudonné.

The theorem states: Let $f\ :\ U\rightarrow \mathbb {R} _{0}^{+}$ 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.