  Modifying the binary inverse function in a variety of ways, like swapping two
output points has been known to produce a $4$-differential uniform permutation
function. Recently, in \cite{Li19} it was shown that this swapped version of
the inverse function has boomerang uniformity exactly $10$, if $n\equiv 0\pmod
6$, $8$, if $n\equiv 3\pmod 6$, and 6, if $n\not\equiv 0\pmod 3$. Based upon
the $c$-differential notion we defined in \cite{EFRST20} and $c$-boomerang
uniformity from \cite{S20}, in this paper we characterize the $c$-differential
and $c$-boomerang uniformity for the $(0,1)$-swapped inverse function in
characteristic~$2$: we show that for all~$c\neq 1$, the $c$-differential
uniformity is upper bounded by~$4$ and the $c$-boomerang uniformity by~$5$ with
both bounds being attained for~$n\geq 4$.
