Let $X$ and $Y$ be Banach spaces.
Definition: A function $f:X\rightarrow Y$ is called Hadamard differentiable at $x\in X$ tangentially to $U\subseteq X$ iff $x\in U$ and there exists a continuous linear function $f'_x:U\rightarrow Y$ such that for all $u\in U$ it holds that $$\left\Vert\frac{f(x + \delta_nu_n)-f(x)}{\delta_n} - f'_x(u)\right\Vert_Y\rightarrow 0$$ as $n\rightarrow\infty$ for every sequence $(u_n)_{n\in\mathbb N}$ in $X$ converging to $u$ and every sequence $(\delta_n)_{n\in\mathbb N}$ in $\mathbb R$ converging to $0$.
Problem: In my case, $X = \textit{BV}([a,b])\times\textit{BV}([a,b])$, where $\textit{BV}([a,b])$ is the set of all real-valued functions on $[a,b]$ that are of bounded variation, $Y = \mathbb R$, and $$f(F,G) = \int_a^b F\,\mathrm dG.$$ Here the integral is the Lebesgue-Stieltjes integral; note that since real-valued functions on a compact set that are of bounded variation are also bounded and real-valued functions are measurable, $f$ is well-defined. Is $f$ Hadamard differentiable (maybe tangentially to some subset $U$) for each $(F,G)\in X$?
I know that if $X = \big(\textit{BV}([a,b])\cap\textit{D}([a,b])\big)\times\big(\textit{BV}([a,b])\cap\textit{D}([a,b])\big)$, where $D([a,b])$ is the space of real-valued cadlag functions on $[a,b]$, then $f$ is Hadamard differentiable with derivative $$f_{(F,G)}'(A,B) = B(b)F(b) - B(a)F(a) - \int_a^b B_{-}\,\mathrm dF + \int_a^b A\,\mathrm dG,$$ where $B_{-}$ is the left-continuous version of the right-continuous function $B$. <Edit: In an earlier version of this post, there was an error. I wrote $B(F(a))$ instead of $B(a)F(a)$ (and similar for $b$). The error is also in the comments. I hope this error didn't cause any confusion.>
However, since I am operating on the larger space $\textit{BV}([a,b])\times\textit{BV}([a,b])$, I suppose that this result no longer holds. Therefore my question: is there any hope that $f$ is still Hadamard differentiable on this larger space? Unfortunately, I have no idea how to tackle this problem. So any help is appreciated.
The functions I want apply $f$ to are the of the form $\frac{L + R}2$, where $L$ and $R$ are left- and right-continuous functions with the same points discontinuities (i.e., if $L$ is discontinuous at $t\in[a,b]$, then so is $R$). So I have some form of continuity and an at most countable set of discontinuities. Maybe that's enough to have Hadamard differentiability tangentially to the set of these functions?