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.

Mon May 5 12:53:33 CDT 1997