Editing Symmetry of second derivatives (section)

== Requirement of continuity ==
The symmetry may be broken if the function fails to have differentiable partial derivatives, which is possible if Clairaut's theorem is not satisfied (the second partial derivatives are not [[Continuous function|continuous]]).

[[File:Graph001.png|thumb|right|The function ''f''(''x'',&thinsp;''y''), as shown in equation ({{EquationNote|1}}), does not have symmetric second derivatives at its origin.]]
An example of non-symmetry is the function (due to [[Peano]]){{sfn|Hobson|1921|pages=403–404}}{{sfn|Apostol|1974|pages=358–359}}
{{NumBlk|:
| <math>f(x,\, y) = \begin{cases}
  \frac{xy\left(x^2 - y^2\right)}{x^2 + y^2} & \mbox{ for } (x,\, y) \ne (0,\, 0),\\
                                           0 & \mbox{ for } (x,\, y) = (0,\, 0).
\end{cases}</math>
| {{EquationRef|1}}
}}

This can be visualized by the polar form <math>f(r \cos(\theta), r\sin(\theta)) = \frac{r^2 \sin(4\theta)}{4}</math>; it is everywhere continuous, but its derivatives at {{nowrap|(0,&thinsp;0)}} cannot be computed algebraically. Rather, the limit of difference quotients shows that <math>f_x(0,0) = f_y(0,0) = 0</math>, so the graph <math>z = f(x, y)</math> has a horizontal tangent plane at {{nowrap|(0,&thinsp;0)}}, and the partial derivatives <math>f_x, f_y</math> exist and are everywhere continuous. However, the second partial derivatives are not continuous at {{nowrap|(0,&thinsp;0)}}, and the symmetry fails. In fact, along the ''x''-axis the ''y''-derivative is <math>f_y(x,0) = x</math>, and so:
:<math>
  f_{yx}(0,0) =
  \lim_{\varepsilon \to 0} \frac{f_y(\varepsilon,0) - f_y(0,0)}{\varepsilon} =
  1.
</math>

In contrast, along the ''y''-axis the ''x''-derivative <math>f_x(0,y) = -y</math>, and so <math>f_{xy}(0,0) = -1</math>. That is, <math>f_{yx} \ne f_{xy}</math> at {{nowrap|(0,&thinsp;0)}}, although the mixed partial derivatives do exist, and at every other point the symmetry does hold.

The above function, written in polar coordinates, can be expressed as
:<math>f(r,\, \theta) = \frac{r^2 \sin{4\theta}}{4},</math>

showing that the function oscillates four times when traveling once around an arbitrarily small loop containing the origin. Intuitively, therefore, the local behavior of the function at (0,&thinsp;0) cannot be described as a quadratic form, and the Hessian matrix thus fails to be symmetric.

In general, the [[interchange of limiting operations]] need not [[commutative property|commute]]. Given two variables near {{nowrap|(0,&thinsp;0)}} and two limiting processes on
:<math>f(h,\, k) - f(h,\, 0) - f(0,\, k) + f(0,\, 0)</math>

corresponding to making ''h'' → 0 first, and to making ''k'' → 0 first. It can matter, looking at the first-order terms, which is applied first. This leads to the construction of [[Pathological (mathematics)|pathological]] examples in which second derivatives are non-symmetric. This kind of example belongs to the theory of [[real analysis]] where the pointwise value of functions matters.  When viewed as a distribution the second partial derivative's values can be changed at an arbitrary set of points as long as this has [[Lebesgue measure]] 0. Since in the example the Hessian is symmetric everywhere except {{nowrap|(0,&thinsp;0)}}, there is no contradiction with the fact that the Hessian, viewed as a [[Schwartz distribution]], is symmetric.