For the part highlighted in green I assume we are using Theorem 5.11 (below). However, from the the calculations of the partial derivatives we know that $\frac{\partial g}{\partial x}$ and $\frac{\partial g}{\partial y}$ exist and are equal to $0$ for $(x,y)=(0,0)$ and hence continuous at $(x,y)=(0,0)$.

But the above implies $g$ is differentiable at $(0,0)$. Where am I going wrong?

![enter image description here](https://i.stack.imgur.com/rgJnU.jpg)

![enter image description here](https://i.stack.imgur.com/5ajo1.jpg)

![enter image description here](https://i.stack.imgur.com/3moJu.jpg)