# Short question about continuity of derivative

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.

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