Assume is a real-valued function defined on an open set of and all partial derivatives of exist on . Is it possible that there is some and such that the directional derivative of in the direction of at does not exist?

Thanks!

- February 23rd 2010, 07:41 PMzzzhhhA question about partial derivatives and directional derivative.
Assume is a real-valued function defined on an open set of and all partial derivatives of exist on . Is it possible that there is some and such that the directional derivative of in the direction of at does not exist?

Thanks! - February 24th 2010, 03:10 AMOpalg
- February 24th 2010, 04:05 AMHallsofIvy
If f is

**differentiable**at a point, then the directional derivative in all directions exists. But, as Opalg showed, it may happen that has partial derivatives at a point but is not "differentiable" there.

Another example is

At (0,0) both partial derivatives exist and are 0. But the function is not differentiable and directional derivatives do not exist except in the directions of the axes. - February 24th 2010, 08:17 AMOpalg
Umm, I hate to disagree, but that example does not have partial derivatives at the origin:

(ouch!) (Angry)

I think you probably meant But no, on second thoughts that won't work either, because that function doesn't have partial derivatives on either of the axes (except at the origin). I have a feeling that the example I gave previously can't really be simplified. - February 24th 2010, 02:42 PMzzzhhh
Thank you! Now I have a further question: Assume the conditions in my first post still hold, that is, is a real-valued function defined on an open set of and all partial derivatives of exist on . If the directional derivative of in the direction of at a point does exist, is it always true that the value of the directional derivative equals the inner product ? Thanks!

- February 24th 2010, 02:54 PMhjortur
- February 25th 2010, 01:01 AMOpalg
- February 25th 2010, 04:20 AMHallsofIvy
(Crying) Ouch! You are right. That is the example I meant to give. It does have partial derivatives at (0,0) but is not differentiable there. exists at (x, 0) but not and vice-versa at (0, y).

That is not a good example because the initial post asked for an example where the partial derivatives exist on some**open set**. Thank you. - February 25th 2010, 05:05 PMzzzhhh