Editing Symmetry of second derivatives (section)

== Sufficiency of twice-differentiability ==
A weaker condition than the continuity of second partial derivatives (which is implied by the latter) which suffices to ensure symmetry is that all partial derivatives are themselves [[Differentiable function#Differentiability in higher dimensions|differentiable]].{{sfn|Hubbard|Hubbard|2015|pages=732–733}} Another strengthening of the theorem, in which ''existence'' of the permuted mixed partial is asserted, was provided by Peano in a short 1890 note on [[Mathesis (journal)|Mathesis]]:

: ''If <math>f:E \to \mathbb{R}</math> is defined on an open set <math>E \subset \R^2</math>; <math>\partial_1 f(x,\, y)</math> and <math> \partial_{2,1}f(x,\, y)</math> exist everywhere on <math>E</math>; <math>\partial_{2,1}f</math> is continuous at <math>\left(x_0,\, y_0\right) \in E</math>, and if <math>\partial_{2}f(x,\, y_0)</math> exists in a neighborhood of <math>x = x_0</math>, then <math>\partial_{1,2}f</math> exists at <math>\left(x_0,\, y_0\right)</math> and <math>\partial_{1,2}f\left(x_0,\, y_0\right) = \partial_{2,1}f\left(x_0,\, y_0\right)</math>.''{{sfn|Rudin|1976|pages=235–236}}