Past exam of the mathematics course of the University of Cambridge / 2004 / ia / Paper 3 / 9C (Vector Calculus)

Set $r=\sqrt{x^2+y^2}$. The bound $|xy|\le r^2/2$ and $|x^2-y^2|\le r^2$ give $|f(x,y)|\le r^2/2$. Hence $f\to 0$ at the origin, proving continuity, and $|f(x,y)|/r\to 0$, proving Fréchet differentiability there with derivative equal to the zero linear map.

Along the coordinate axes, direct difference quotients giveThese formulas include the origin. Therefore the two mixed partial derivatives differ:The explicit operator order removes any ambiguity about the notation for mixed derivatives. Also $f_{xx}(0,0)=f_{yy}(0,0)=0$, since the first derivatives restricted to their own axes vanish identically.

All four second-order partial derivatives are discontinuous at the origin. Away from it,Along $x=y\ne 0$ these are one and minus one, rather than their zero values at the origin. Meanwhile ${\partial }_y{\partial }_{x}f(x,0)=1$ for $x\ne 0$, whereas its value at the origin is minus one; and ${\partial }_x{\partial }_{y}f(0,y)=-1$ for $y\ne 0$, whereas its origin value is one. This gives a concrete unequal mixed partial derivatives example despite differentiability of the original function.

True. Differentiability implies continuity.

False. Partial derivatives do not imply continuity; derivatives along coordinate axes need not control nearby non-axis approaches.

False. Directional derivatives do not imply differentiability; the resulting dependence on direction need not be a linear derivative.

False. Differentiability does not imply continuous derivatives.

False. Unequal mixed partial derivatives can exist when the continuity hypotheses of Clairaut's theorem fail.