Unequal mixed partial derivatives
= Unequal mixed partial derivatives
{title2=$\partial_y\partial_xf(0)=-1,\quad\partial_x\partial_yf(0)=1$}
For $f(x,y)=xy(x^2-y^2)/(x^2+y^2)$ away from the origin and zero at the origin, $|f|\leq(x^2+y^2)/2$ proves <Fréchet differentiability> with zero derivative. Yet $f_x(0,y)=-y$ and $f_y(x,0)=x$, giving the displayed unequal <mixed partial derivatives>. They exist but are not continuous at the origin, so <Clairaut's theorem> does not apply.