What are the conditions for a function $\phi(x,y)$ to have directional derivative at a point $(a,b)$ but $\phi$ is not differentiable at $(a,b)$?

Question

What are the conditions for a function $\phi(x,y)$ to have directional derivative at a point $(a,b)$ but $\phi$ is not differentiable at $(a,b)$?

Answer:

In my opinion, they are:

1. $\phi$ is not continuous at $(a,b)$ $\implies$ $\phi$ is not differentiable at $(a,b)$
2. $\phi$ is continuous at $(a,b)$, $\phi_x,\phi_y$ exist but $\phi_x,\phi_y$ are not continuous $\implies$ $\phi$ is not differentiable at $(a,b)$

My questions

1. Is the above claim is correct and complete? Is there any other case(s) left?
2. Please suggest a book where the arguments are clearly mentioned

Thanks a lot in advance.

Answer 1

The claim $2$ is generally wrong. A function can be differentiable with non-continuous partial derivatives. The following link provides an example : https://mathinsight.org/differentiable_function_discontinuous_partial_derivatives

You can only say that if the partial derivatives are continuous, then the function is differentiable. However you cannot conclude anything if the partial derivatives are not continuous.