Lets say we have the scalar field
So by inspection, it seems that is continuous at . Now the partial derivatives of are not continuous at . So is not differentiable at . So this is not Fréchet differentiable? But could it be Gâteaux differentiable?
Lets say we have the scalar field
So by inspection, it seems that is continuous at . Now the partial derivatives of are not continuous at . So is not differentiable at . So this is not Fréchet differentiable? But could it be Gâteaux differentiable?
The Frechet derivative is for functions , and it is a matrix (or a coloumn or row matrix). If it exists it is expressed in terms of its partial derivatives. However, existence of its partial derivatives does not gaurenntee Frechet differenciablility. There is a theorem which says if the partial derivatives are continous then the function is differenciable. I am not sure about the converse. I think it is wrong. You need to justify non-Frechet differenciability in another way. I am not familar with Gateaux derivative.
Note: The function is continous because