I want to ask a question that I feel everyone acts as if it is a common knowledge. I am really new to functional analysis. How does Fréchet differentiability imply continuity? In the definition I have only continuity of the derivative itself,A, is mentioned:
A mapping F : U ⊂ U → V is said to be Fréchet differentiable at u ∈ U if there exist an operator A ∈ L(U,V ) and a mapping r(u,·) : U → V with the following properties: for all h ∈ U such that u + h ∈ U, we have F(u + h) = F(u) + Ah + r(u,h) where the so-called remainder r satisfies the condition $\frac{||r(u,h)||_V}{||h||_U} $ → $0$ as $|h||_U$ → 0. The operator A is then called the Fréchet derivative of F at u, and we write A = F ? (u). If A is Fréchet differentiable at every point u ∈ U, then A is said to be Fréchet differentiable in U.
$F(u+h)-F(u) \to 0$ as $h \to 0$ because $Ah \to 0$ as $h \to 0 $ and $r(u,h) =\|h\|_U \frac {r(u,h)} {\|h\|_U} \to 0$ too. [$\|Ah\| \leq \|A\| \|h\|_U \to 0$].