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 $β_i$ (a basis of $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$), some functions $f_i$ of $c$-differential uniformities $δ_i$, and $L_i$ (specific linearized polynomials defined in terms of $β_i$), $1\leq i\leq n$, then $F(x)=\sum_{i=1}^nβ_i f_i(L_i(x))$ has $c$-differential uniformity equal to $\prod_{i=1}^n δ_i$.