If $f_x(0,0)$ doesn't exists, can we conclude that $f_{xy}(0,0)$ doesn't exist?

Asked Sep 01 '16 at 05:43

Active Sep 01 '16 at 05:52

Viewed 36 times

Given that $f:\mathbb{R}^2 \rightarrow \mathbb{R}$. Suppose we have $f_x$ is defined for all points $(x,y)$ except $(0,0)$.

Is it possible for $f_{xy}(0,0)$ to exist?

I think it is not possible by using definition of partial derivative.

Remark: By $f_{xy}$, I mean differentiate $f$ w.r.t. $x$ first, followed by $y$.

edited Sep 01 '16 at 05:52

asked Sep 01 '16 at 05:43

Idonknow

2

If you mean first differentiate wrt to y and then to x, then I'd say that is possible for it to exists. If you mean first differentiate wrt to x and then to y, the I'd say no since you can't differentiate there if the first derivative doesn't exists – Ant Sep 01 '16 at 05:49
I too believe it's not possible. – babakks Sep 01 '16 at 06:57

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