A function 
is Fréchet differentiable at 
if
 |
exists. This is equivalent to the statement that 
has a removable discontinuity
at 
,
where
 |
In literature, the Fréchet derivative is sometimes known as the strong derivative (Ostaszewski 2012) and can be seen as a generalization of the gradient to arbitrary vector spaces (Long 2009).
Every function which is Fréchet differentiable is both Carathéodory differentiable and Gâteaux differentiable. The relationship between the Fréchet
derivative and the Gâteaux derivative
can be made even more explicit by noting that a function 
is Fréchet differentiable if and only if the limit
used to describe the Gâteaux derivative exists uniformly with respect to vectors
on the unit sphere of the domain space 
; as such, this uniform limit (when it exists) is what's called
the Fréchet Derivative (Andrews and Hopper 2011)
