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Let $X$ be a Banach space and $\Omega \subset X$ be open.

The functional $f$ has a Gâteaux derivative $g \in X'$ at $u \in \Omega$ if, $\forall h\in X,$ $$\lim_{t \rightarrow 0}[f(u+th)-f(u)- \langle g,th \rangle]=0$$

How can I prove the following:

If $f$ has a continuous Gâteaux derivative on $\Omega$, then $f \in C^1(\Omega,\mathbb R)$.

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    How do you define $C^1$?2012-05-16
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    You need to divide the term in the limit by $t$. Any continuous function satisfies the condition above.2012-05-16
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    The Gâteaux differential is usually defined as $df(x;h) = \lim_{t\rightarrow 0} \frac{f(x+t h) - f(x)}{t}$. To show that it is a Fréchet derivative, you need to show linearity and continuity in $h$.2012-05-16
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    @copper.hat That depends on what you are used to. For me the Gâteaux derivative is a bounded linear functional, by definition. "Some authors, such as Tikhomirov (2001), draw a further distinction between the Gâteaux differential (which may be nonlinear) and the Gâteaux derivative (which they take to be linear)." -- wikipedia.2012-05-16
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    @LeonidKovalev: You are right, I was a bit quick off the mark. I had forgotten the distinction between Gâteaux differential and derivative. The formula still needs to be divided by $t$.2012-05-16

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