Let .

Demonstrate that has directional derivatives in all directions in but that is not differentiable in .

My attempt :

I notice I can't use the theorem that states that if is differentiable in a point, then it has directional derivatives in all directions in this point.

I believe is continuous in ... I didn't prove it via the definition but I've failed at showing that it is not continuous. I realize that it would have been easy if wasn't continuous because could not be differentiable but this is not the case.

I'm stuck at starting the exercise.

I guess I have to use the definition to show that is not differentiable, i.e. that there does not exist a linear function such that .

Seems quite hard, I don't know how to start.