  In this paper we focus on the relation between Riemann integrability and weak
continuity. A Banach space $X$ is said to have the weak Lebesgue property if
every Riemann integrable function from $[0,1]$ into $X$ is weakly continuous
almost everywhere. We prove that the weak Lebesgue property is stable under
$\ell_1$-sums and obtain new examples of Banach spaces with and without this
property. Furthermore, we characterize Dunford-Pettis operators in terms of
Riemann integrability and provide a quantitative result about the size of the
set of $\tau$-continuous non Riemann integrable functions, with $\tau$ a
locally convex topology weaker than the norm topology.
