We defined that a function is Gateaux differentiable, if all directional derivatives exist. I just wanted to check, whether I got a few things right: Now I wanted to ask, whether it is true that if $(x_0,h) \mapsto d(x_0,h)$ is continuous, then $df(x_0,.)$ is linear?
and if $df(x_0,.)$ is bounded and linear and $x_0 \mapsto d(x_0,.)$ is continuous, then $f$ is continuously Frechet-differentiable.
I am just asking this, because from this, we would have that if $(x_0,h) \mapsto d(x_0,h)$ is continuous then $f$ is continuously Frechet-differentiable and I am not sure whether this is true.
I think this wikipedia article covers some of the questions I am asking, but maybe you could recheck whether every I said is true? http://en.wikipedia.org/wiki/G%C3%A2teaux_derivative#Linearity_and_continuity