In Clairaut's theorem based on equality of mixed second order partial derivatives, in some books the hypothesis is: if $f_{xy}$ and $f_{yx}$ are defined in a disk around $(x_o,y_o)$ and they are continuous at $(x_o,y_o)$, then $f_{xy}(x_o,y_o)=f_{yx}(x_o,y_o)$. While some books say that if $f_{xy}$ and $f_{yx}$ are continuous throughout an open disk around $(x_o,y_o)$, then $f_{xy}(x_o,y_o)=f_{yx}(x_o,y_o)$. Can anyone suggest that whether the continuity of the two mixed partials at the point enough to conclude the equality of the two. I have gone through the proof but towards the last they just say that the iterated limits are equal and hence $f_{xy}(x_o,y_o)=f_{yx}(x_o,y_o).$
Please explain.
Thanks.
