Is non-differentiable having the derivative a contradiction?

Question

I'm reading this book and on page 104 they define what means a function being differentiable:

Afterwards they give the following example:

Is it not a contradiction? Following the definition $f$ is differentiable at $a$ if there is $D_f(a)$. In the example, the derivative $D_f(0,0)$ does exist at $(0,0)$. What am I missing?

Answer 1

You are correct; the final sentence in the second excerpt is simply an error. The derivative $D_f(0,0)$ does not exist, nor does the gradient $\nabla f(0,0)$ (as defined earlier on page 110). All that is true is that the partial derivatives exist.

Answer 2

I think the book is right. Even though all the partial derivatives exist, since the function is not continuous at (0,0), the function can't be Frechet differentiable at that point.