Denote by \(d x\) one-dimensional Lebesgue measure and let \(f \in \mathcal{L}^{1}(d x)\). Define the mean value

\[M_{h} f(x):=\frac{1}{2 h} \int_{x-h}^{x+h} f(t) d t\]

and show that

(i) \(M_{h} f(x)\) is continuous and \(\left\|M_{h} fight\|_{1} \leqslant\|f\|_{1}\),

(ii) \(\lim _{h ightarrow 0}\left\|M_{h} f-fight\|_{1}=0\).