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

The theorem states: Let [math]\displaystyle{ f\ :\ U \rightarrow \R^{+}_0 }[/math] be a function of class [math]\displaystyle{ C^{2} }[/math] in an open set U contained in [math]\displaystyle{ \R^n }[/math], then [math]\displaystyle{ \sqrt{f} }[/math] is of class [math]\displaystyle{ C^{1} }[/math] in U if and only if its partial derivatives of first and second order vanish in the zeros of f.