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A function whose mixed partials are unequal

We get so used to the fact that tex2html_wrap_inline637 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 tex2html_wrap_inline639 continuous in a neighborhood of a point. Then tex2html_wrap_inline641 will exist and equal tex2html_wrap_inline639 in that neighborhood.

Functions whose second partials are discontinuous need not have their mixed partials equal. The standard example is

displaymath645

Then

displaymath647

where tex2html_wrap_inline649 must be computed separately by using tex2html_wrap_inline651 . Similarly,

displaymath653

Now, tex2html_wrap_inline655 , which is tex2html_wrap_inline657 , and tex2html_wrap_inline659 , which is tex2html_wrap_inline661 . So, for this function tex2html_wrap_inline663 .

Here's a picture of f(x,y). The interesting part is that it doesn't look that strange.

external



Tom Vogel
Mon May 5 12:53:33 CDT 1997