# Definition talk:Differentiable Functional

I don't know if the book in question gives one, but would you mind figuring out a formal definition? It seems like you'd need to do something like replace the $\epsilon$ by a function and requiring that it tends to zero with $h$, or something. Depends on what $\Delta$ means.
Also: is the convergence $\epsilon\to0$ uniform in $y$? I guess not, because that would be strong. Locally uniform maybe? --barto (talk) 14:53, 30 April 2017 (EDT)
$\Delta J$ is defined here. As for the type of convergence, it is not mentioned. At most I am assuming the limit $\displaystyle \lim_{\size h \mathop \to 0} \epsilon = 0$, where $\size h$ is a norm in function space, and in this case is not defined. The norm itself is described in the book, but for this definition it is not clear which norm he has in mind. Probably the one, which suits the problem in question the best. The author does not consider dependence of $\epsilon$ on $y$ to a great extent. All he wants that the second term would be just slightly steeper than linear in $\size h$.