Suppose $f:X \to Y$ is map between Banach spaces.
We know that the Frechet derivative of $f$ at $x$ is a bounded map by definition if it exists (satisfying certain properties that I won't write) $f'(x):X \to Y$.
We also know that if one can show that the Gateaux derivative of $f$ at $x$, $DF(x):X \to Y$ is continuous at $x$, then the Frechet derivative at $x$ exists and is given by the Gateaux derivative at $x$.
I don't think that the Gateaux derivative of any function is required to be bounded. I am reading the free book Applied Analysis by Hunter and it doesn't say anything, but I am not sure.
So my question is, if I find the Gateaux derivative of a function at $x$ and show that it is continuous at $x$ without showing that it is bounded (i.e., $\lVert DF(x)h\rVert \leq C\lVert h\rVert$), can I still use the theorem to say that the Frechet derivative exists and hence is bounded? It seems wrong to say so.
By what I wrote above, the answer should be yes. But I have seen a lot of conflicting statements about Gateaux derivatives so I'm not sure.