I dont understand this statement: Prove that continuity in $x$ of the Gateaux derivative implies Frechet differentiability

Question

Prove that continuity in $x$ of the Gateaux derivative implies Frechet differentiability

I don't understand this statement, since Gateaux derivative is a function $f(x;y)=a\cdot y$ for all $y$, defined in the point $x$ (not variable) and its variable is $y$, so I don't get what continuity in $x$ means.

Answer 1

Well, first when it was defined, $x$ was assumed not to vary. On the other hand, I guess, it was also assumed to be arbitrary (in some domain).

So, $f(x;y)$ is defined in all $x$'s, thus later, we can do vary that point $x$.