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Suppose I have a map $F:X \to Y$ where X and Y are Banach spaces of continuously differentiable functions (of some order). Let $F(f) = \partial_tf(x,t) + \frac{f(x,t)}{(1-g(x,t)f(x,t))^2}$

Let $f_0$ be a particular point in $X$. How does one go about showing that $F$ is Gateaux differentiable in an open subset of $f_0$? Clearly $F(f)$ is G-differentiable with the constraint that $1\neq g(x,t)f(x,t)$, right? But this depends on $f$ itself, so how can I show that it's true in some open subset of $f_0$?

Let $f_0 = g(x,t)t$ if that helps.

Basically I am confused about what functions are in the open set and how can I say that "as long as the denominator isn't zero $F$ is differentiable".

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    You don't know that: it is false. What is true is that everything in an open $\varepsilon$-neighborhood of $f$ is of the form $f + \tilde{f}$ where $\lvert \tilde{f}\rvert \le \varepsilon$. If $\varepsilon$ is sufficiently small then $g(f+\tilde{f})$ will stay away from $1$.2012-08-22

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