It is a general fact that the notion of Gateaux derivative is not uniform over the mathematical community i.e. someone requires it to be linear and continue other not and require this additional property explicitly if they have to use it. Now I'm referring to the book Drabek, Milnor - Methods of Nonlinear Analysis
as you can see in the example 3.2.3 (I don't post the second page because it's obvious) Drabek, Milota build the Jacobi Matrix of the Gateaux Derivative of a general function f. BUT not every function has a linear Gateaux derivative (a matrix is the representation of a linear - and in this case continuous- function right?)! (think of $x \to |x|$ for example). Is there something I'm missing?