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,{{cite journal

| last1=Glaeser | first1=Georges | authorlink1=Georges Glaeser

| title=Racine carrée d'une fonction différentiable

| journal=Annales de l'Institut Fourier

| volume=13

| issue=2

| date=1963

| pages=203–210

| doi=10.5802/aif.146 | doi-access=free | url=http://www.numdam.org/item?id=AIF_1963__13_2_203_0}} and was later simplified by Jean Dieudonné.{{cite journal

| last1=Dieudonné | first1=Jean | authorlink1=Jean Dieudonné

| title=Sur un théorème de Glaeser

| journal=Journal d'Analyse Mathématique

| volume=23

| date=1970

| pages=85–88

| zbl=0208.07503

| doi=10.1007/BF02795491 | doi-access=free}}

The theorem states: Let $f\backslash \; :\backslash \; U\; \backslash rightarrow\; \backslash R^\{+\}\_0$ be a function of class $C^\{2\}$ in an open set U contained in $\backslash R^n$, then $\backslash sqrt\{f\}$ 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.