Glaeser's continuity theorem

Short description: 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 $C^2$. It was introduced in 1963 by Georges Glaeser, and was later simplified by Jean Dieudonné.

The theorem states: Let $f:U\to {\mathbb{ℝ}}_0^{+}$ 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.