  In this paper, we construct some piecewise defined functions, and study their
$c$-differential uniformity. As a by-product, we improve upon several prior
results. Further, we look at concatenations of functions with low differential
uniformity and show several results. For example, we prove that given $\beta_i$
(a basis of $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$), some functions $f_i$ of
$c$-differential uniformities $\delta_i$, and $L_i$ (specific linearized
polynomials defined in terms of $\beta_i$), $1\leq i\leq n$, then
$F(x)=\sum_{i=1}^n\beta_i f_i(L_i(x))$ has $c$-differential uniformity equal to
$\prod_{i=1}^n \delta_i$.
