The Frechet derivative is for functions , and it is a matrix (or a coloumn or row matrix).Ifit 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