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I am wondering about a problem from the calculus of derivatives in Banachspaces. It is about the difference between two definitions of continuity concerning the Gâteaux-Derivative of a function $P:X\rightarrow Y$ in Banach spaces $X,Y$.The Gâteaux-Derivative in $f$ in direction $h$ is then defined as: \begin{align} DP(f)h=lim_{t \rightarrow 0}[P(f+th)-P(f)]/t \end{align} There are now two different ways in defining the continuity of $DP$. The first way is that $DP$ as a function in the two variables $f$ and $h$ is continuous on the Product-Space: \begin{equation} DP:(U\subset X)\times X \rightarrow Y \ \ \ \ \ (1) \end{equation} is continuous. The second way is two consider $DP$ as a map from the Banach space $X$ or better a subset $U \subset X$ to the space $L(X,Y)$ of the linear functionals from $X$ to $Y$: \begin{equation} DP:U\rightarrow L(X,Y) \ \ \ \ \ \ \ \ \ \ \ \ \ (2) \end{equation} is continuous. The question now is if these two definitions are equivalent in the case of Banach spaces. In the case of Fréchet-Spaces $X,Y$ its obviously not equivalent because in general $L(X;Y)$ is not a Fréchet-Space anymore, its not metrizable in general . But what about the Banach spaces? Is it equivalent in this case? What we know is that when $P$ is Gâteaux-differentiable and $DP$ continuous,in the sense of continuity on product spaces, then $P$ is also Fréchet-differentiable and the Fréchet-Derivative is the same as the Gâteaux-Derivative. As discussed for example in Gâteaux derivative. But is it also continuous in the sense of continuity of Fréchet-Derivatives, continuous as a map described in (2). Does anybody maybe know a counterexample?

I think that an example for a function, gateux-differentiable and continous in the first sense on the product space $U\times Y$, but not continuous as a function $U\rightarrow L(X,Y)$ should be the function $P:L^{1}([0,\pi])\rightarrow \mathbb{R}$ given by \begin{equation} f\rightarrow \int\limits_{0}^{\pi} \sin(f(t))dt \end{equation} This function is described in, http://www.m-hikari.com/ams/ams-password-2009/ams-password17-20-2009/gaxiolaAMS17-20-2009.pdf, where it is shown that $P$ is gateux-differentiable but not frechet-differentiable. It is not continuous in the second sense, which one can easily see by testing with functions centered on a small Intervall with values $n \in \mathbf{N}$. But it should be continuous on the product space $U\times X$.

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  • $\begingroup$ Just quick comment: it shouldn't be; because directional derivatives does not imply continuity of derivative or even the existence of the latter: consider $f(\xi_1, \xi_2) = \dfrac{\xi_1 \xi_2}{\xi_1^4 + \xi_2^2}.$ Unless I quite didn't get your question, the existence of derivative implies the existence of directional derivatives; if the former is continuous (as a function), the latters are also continuous (sum of compositions by inclusions and projections). But maybe I am missing something. $\endgroup$ – Will M. Jan 16 '17 at 17:41
  • $\begingroup$ There is something odd in this statement: " What we know is that when $P$ is Gâteaux-differentiable and $DP$ continuous,in the sense of continuity on product spaces, then $P$ is also Fréchet-differentiable and the Fréchet-Derivative is the same as the Gâteaux-Derivative" If $f$ is defined in the product of Banach spaces, then the existence of partials everywhere in some ball and continuity in the centre of the ball imply the existence of the the derivative. $\endgroup$ – Will M. Jan 18 '17 at 22:31
  • $\begingroup$ I am writing a book in the topic of differential calculus (in normed vector spaces, either complete or not) and I provide an example of a function whose partials exist and are continuous and fail to be differentiable. The example is $f(\xi_1, \xi_2) = \xi_1^{\frac{2}{3}}$ for $\xi_1 = \xi_2 \geq 0$ and $f(\xi_1, \xi_2) = 0$ elsewhere. Partials exist on the set $\xi_1 \neq \xi_2$ and at $(0,0),$ are continuous at $(0,0)$ but $f$ is not differentiable. $\endgroup$ – Will M. Jan 18 '17 at 22:35
  • $\begingroup$ You are quite right that this is odd . If $P$ is gateux-differentiable, then we need $DP$ to be continuous as a function \begin{equation} DP:U\rightarrow L(X,Y)\end{equation}. See for example(Prop. 3.2.15 in Drabek--Milota); $\endgroup$ – Varg Veum Jan 20 '17 at 9:44

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