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.