For $h$ to be differentiable at the origin, $\mathrm{\Δ} h\equiv h\left(0\+h\,0\+k\right)-h\left(0\,0\right)\=h\left(h\,k\right)$ must assume the form

$h_x\left(0\,0\right) h\+h_y\left(0\,0\right) k\+\mathrm{\η}$$\left(h\,k\right)\cdot \sqrt{h^2\+k^2}$ 

where $\mathrm{\η}\to 0$ as $\left(h\,k\right)\to \left(0\,0\right)$. Since $h_x\left(0\,0\right)\=h_y\left(0\,0\right)\=0$ from Example 4.11.7, it follows that

where $\mathrm{\λ}\left(x\,y\right)\=\frac{hk \left(h^2-k^2\right)}{{\left(h^2\+k^2\right)}^{3\/2}}$. To show that $\mathrm{\λ}\to 0$ as $\left(h\,k\right)\to \left(0\,0\right)$, make the following estimate.

where Inequalities 4 and 5 from Table 3.2.1 have been invoked. Hence, setting $\mathrm{\η}\=\mathrm{\λ}$ implies that $h$ is differentiable at the origin.$$