I think that you even have this in the classical sense - a function with directional derivatives at a point may not be differentiable there.
One defines .
is defined on the maximal subregion .
Now let and .
How to prove, that the following implication is not true in general:
In a book about functional analysis I read the following lemma:
"Let . If the inequality holds for all and for all then is weakly differentiable in with ."
Here the problem is , so the lemma can not be used.
Do you have a counter example?