Past exam of the mathematical methods II course of the University of Oxford / 2015 resit / 3 / Solution

Work on a domain with $x\ne 0$, and suppose $f$ is differentiable at $u=y/x$. The multivariable chain rule givesMultiplying the first equation by $x$ and the second by $y$ cancels their common terms:The geometry is that $y/x$ stays fixed along a radial dilation $(x,y)\mapsto (\lambda x,\lambda y)$. Thus $w$ is constant along each admissible radial ray, and the differential operator $x{\partial }_x+y{\partial }_y$ differentiates along that ray. This is the degree-zero case of Euler homogeneous function theorem. The displayed quotient does not define a function at $x=0$ without additional extension data.