If $f_x:=\partial f/\partial x$ and $f_{xy}:=\partial f_x/\partial y$ exist, and $f_{xy}$ is continuous, does this imply that $f_x$ is also continuous?

I'm not sure if existence and continuity of one of the second partial derivatives imply the continuity first order partial derivative. Thanks.

  • $\begingroup$ $f(x,y)=\lvert x\rvert +xy$, has $f_{xy}$ with perhaps a removable discontinuity, but $f_x$ discontinuous. $\endgroup$ – ziggurism Dec 9 '15 at 5:04
  • $\begingroup$ So the answer is $f_{x}$ is not continuous in this case? $\endgroup$ – EmmaJ Dec 9 '15 at 5:07
  • 1
    $\begingroup$ Yes. $f_x$ is discontinuous, but $f_{xy}$ appears to be continuous. Though it probably is not $\endgroup$ – ziggurism Dec 9 '15 at 5:09

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