We get so used to the fact that 
for reasonable functions that it's
easy to forget that there is a hypothesis to be satisfied. One condition that
will ensure this is to have 
continuous in a neighborhood of a point.
Then 
will exist and equal in that neighborhood.
Functions whose second partials are discontinuous need not have their mixed partials equal. The standard example is
Then
where 
must be computed separately by using 
. Similarly,
Now, 
, which is
, and 
, which is 
. So,
for this function 
.
Here's a picture of f(x,y). The interesting part is that it doesn't look that strange.
