Let \(\phi \in \mathcal{L}^{1}([0,1], d x)\) and define \(f(t):=\int_{[0,1]}|\phi(x)-t| d x\). Show that

(i) \(f\) is continuous,

(ii) \(f\) is differentiable at \(t \in \mathbb{R}\) if, and only if, \(\lambda\{\phi=t\}=0\).