'a.e.' is a tricky business. When working with 'a.e.' properties one has to be extremely careful. For example, the assertions ' \(u\) is continuous a.e.' and ' \(u\) is a.e. equal to an (everywhere) continuous function' are far apart! Illustrate this by considering the functions \(u=\mathbb{1}_{\mathbb{Q}}\) and \(u=\mathbb{1}_{[0, \infty)}\).