Let $\delta$ be a small positive number, and $x\in [0,1]$, $r\in D(B):= [\delta,1-\delta]$. Consider the function:
$$b(x;r)=\begin{cases}1 \quad |x-r|<\delta\\ 0 \quad elsewhere \end{cases}$$
For every $r$, the operator $B(r)$ is in $\mathcal{L}(\mathbb{R},L^2(0,1))$ and maps $u\in \mathbb{R}$ to $b(x;r)u\in L^2(0,1)$. What is the Frechet derivative of $B(r)$ with respect to $r$ as an operator from $D(B)(\subset \mathbb{R})$ to $\mathcal{L}(\mathbb{R},L^2(0,1))$.