Show that every increasing function \(u: \mathbb{R} ightarrow \mathbb{R}\) is \(\mathscr{B}(\mathbb{R}) / \mathscr{B}(\mathbb{R})\)-measurable. Under which additional condition(s) do we have \(\sigma(u)=\mathscr{B}(\mathbb{R})\) ?

[ show that \(\{u<\lambda\}\) is an interval by distinguishing three cases: \(u\) is continuous and strictly increasing when passing the level \(\lambda\); and \(u\) jumps over the level \(\lambda\); and \(u\) is 'flat' at level \(\lambda\). Draw a picture of these situations.]