The following theorem about equality of partial derivatives was known to Euler around 1734 and was proved later.
Theorem 17.9.1 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
 (17.12) 
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 17.9.2 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 17.9.3 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 17.9.4 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}
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 17.9.5 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.