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 $$f\ :\ U \rightarrow \R^{+}_0$$ 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.