Re: Advanced calculus example needed

On Fri, 15 Jun 2007 08:23:28 +0200, Denis Feldmann
<denis.feldmann.asupprimer@xxxxxxxxxxxxxxxx> wrote:

[Mr.] Lynn Kurtz a écrit :
On Fri, 15 Jun 2007 04:27:36 GMT, "TCL" <tlim1@xxxxxxxxxxx> wrote:

Does anyone have an example of a continuous function f: R^2--> R, such that
both partial derivatives exist everywhere, but it is not differentiable at
(0,0)?


Standard example:

f(x,y) = (xy) / ( x^2 + y^2 ) except at (0, 0) and f(0,0) = 0.

It isn't even continuous at (0,0).

Which makes a bad counterexample indeed.

That makes it a very bad counterexample or a very good
one, depending on what the point is: It's very good if
the point is to show just how little follows from
the condition "all partials exist everywhere", very
bad if the point is to show how hard it is to give
conditions that will imply a function is differentiable.

Things like |xy|^a/(x^2+y^2),
with a well chosen, will be better...




--Lynn


************************

David C. Ullrich
.

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