Continuing with the special case where f is defined on an open set in F^{n}, I will next consider an interesting result which was known to Euler in around 1734 about mixed partial derivatives. It was proved by Clairaut some time later. It turns out that the mixed partial derivatives, if continuous will end up being equal. Recall the notation

and

Theorem 5.10.1 Suppose f : U ⊆ F^{2} → ℝ where U is an open set on which f_{x},f_{y}, f_{xy} and f_{yx} exist. Then if f_{xy} and f_{yx} are continuous at the point

Proof: Since U is open, there exists r > 0 such that B
 (5.19) 
Note that

where α,β ∈
If the terms f

Letting

The following is obtained from the above by simply fixing all the variables except for the two of interest.
Corollary 5.10.2 Suppose U is an open subset of X and f : U → ℝ has the property that for two indices, k,l, f_{xk}, f_{xl},f_{xlxk}, and f_{xkxl} exist on U and f_{xkxl} and f_{xlxk} are both continuous at x ∈ U. Then f_{xkxl}
By considering the real and imaginary parts of f in the case where f has values in ℂ you obtain the following corollary.
Corollary 5.10.3 Suppose U is an open subset of F^{n} and f : U → F has the property that for two indices, k,l, f_{xk}, f_{xl},f_{xlxk}, and f_{xkxl} exist on U and f_{xkxl} and f_{xlxk} are both continuous at x ∈ U. Then f_{xkxl}
Finally, by considering the components of f you get the following generalization.
Corollary 5.10.4 Suppose U is an open subset of F^{n} and f : U → F^{m} has the property that for two indices, k,l, f_{xk}, f_{xl},f_{xlxk}, and f_{xkxl} exist on U and f_{xkxl} and f_{xlxk} are both continuous at x ∈ U. Then f_{xkxl}
This can be generalized to functions which have values in a normed linear space, but I plan to stop with what is given above. One way to proceed would be to reduce to a consideration of the coordinate maps and then apply the above. It would even hold in infinite dimensions through the use of the Hahn Banach theorem. The idea is to reduce to the scalar valued case as above.
In addition, it is obvious that for a function of many variables you could pick any pair and say these are equal if they are both continuous.
It is necessary to assume the mixed partial derivatives are continuous in order to assert they are equal. The following is a well known example [2].
Example 5.10.5 Let

From the definition of partial derivatives it follows immediately that f_{x}

Now

while

showing that although the mixed partial derivatives do exist at
Incidentally, the graph of this function appears very innocent. Its fundamental sickness is not apparent. It is like one of those whited sepulchers mentioned in the Bible.