For $k \in \mathbf{N}^*$ and $t \in \mathbf{R}_+$, we set
$$u_k(t) = \int_{k/2}^{(k+1)/2} \frac{tq(u)}{e^{tu}-1} \mathrm{du} \text{ if } t > 0 \text{, and } u_k(t) = \int_{k/2}^{(k+1)/2} \frac{q(u)}{u} \mathrm{du} \text{ if } t = 0.$$
Show that $u_k$ is continuous on $\mathbf{R}$, for all $k \in \mathbf{N}^*$.