Take the 2-minute tour ×  Is showing that $f_{xy}$ is not equal
to $f_{yx}$ at a given point sufficient/good to show that a
function is discontinuous there?  Can you also suggest a clear
method to find without graphing a method to tell whether a
function is continuous or not for a multivariable function?
Further, but unrelated, does $f_{xy}=f_{yx}$ imply in all cases?
I mean is $f_{xxyz}=f_{xyxz}=f_{zyxx}$...?  share|improve this
question add comment  2 Answers  up vote 1 down vote accepted
The answer to your first question is no. There are continuous,
and even continuously differentiable, functions $f(x,y)$
satisfying $f_{xy}(0,0)\neq f_{yx}(0,0).$ In this situation,
Clairaut's theorem ($C^2\Rightarrow f_{xy}=f_{yx}$) implies that
$f_{xy}$ and $f_{yx}$ cannot both be continuous.  For the last
question, notice that Clairaut's theorem applies to $C^2$
functions. If $f$ is $C^4,$ then your equalities will hold.
share|improve this answer I think there are actually 3 questions
up there the OP wants answered. I