Let $X$ and $Y$ be infinite-dimensional Banach spaces over $\mathbb{C}$ and $U\subset X$ be norm open. We say $f:U\rightarrow Y$ is Gateaux differentiable if for each $x\in U$ and each $h\in X$, $$Df(x)[h]:=\lim_{t\rightarrow 0} \frac{f(x+th)-f(x)}{t}$$ exists (where $t$ is taken over the complex numbers).
Certainly, it is not always the case that the map $$U\rightarrow B(X,Y), x\mapsto Df(x)$$ is continuous. However, I am not entirely sure about this if we assume, in addition, that $f$ is locally bounded on $U$.
I am looking for a counter-example to this, assuming it is not true. In the problem I am working on, I am able to show that this map is continuous for a certain special class of functions, but I would like to know if the extra work is really necessary, or if it is just a general fact about working over $\mathbb{C}$ and having local boundedness of $f$.
Note: I have found a reference that claimed, under all these conditions, that the function $f$ is actually Frechet holomorphic. I am unsure if this implies continuity of the above map.