Lower semicontinuity of nonlocal $L^{\mathrm{\infty }}$ energies on $SB{V}_0(I)$

We characterize the lower-semicontinuity of nonlocal one-dimensional energies of the type $$\mathrm{e}\mathrm{s}\mathrm{s}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\underset{(s,t)\in I\times I}{\sup }h([u](s),[u](t)),$$ where $I$ is an open and bounded interval in the real line, $u\in SB{V}_0(I)$ and $[u](r):=u(r^{+})-u(r^{-})$, with $r\in I$.