Exercise 2, § 4

Let $\mu$ be Lebesgue measure on $\mathbf{R}$ and let $\Omega$ be the topological space obtained by equipping the space $\mathscr{L}^1$ with the topology of pointwise convergence in $\mathbf{R}$. Show that the mapping $f \mapsto \int f d\mu$ of $\Omega$ into $\mathbf{R}$ is not continuous at any point of $\Omega$.