Let \(\chi \in \mathcal{C}_{c}^{\infty}(-1,1)\) with \(0 \leqslant \chi \leqslant 1, \chi(0)=1\) and \(\int \chi(x) d x=1\). Set \(\chi_{n}(x):=n \chi(n x)\).

a) Show that supp \(\chi_{n} \subset[-1 / n, 1 / n]\) and \(\int \chi_{n}(x) d x=1\).

b) Let \(f \in \mathcal{C}(\mathbb{R})\) and \(f_{n}:=f \star \chi_{n}\).
Show that \(\left|\partial^{k} f_{n}(x)ight| \leqslant n^{k} \sup _{y \in \mathbb{B}(x, 1 / n)}|f(y)|\left\|\partial^{k} \chiight\|_{L^{1}}\).

c) Let \(f \in \mathcal{C}(\mathbb{R})\) uniformly continuous. Show that \(\lim _{n ightarrow \infty}\left\|f \star \chi_{n}-fight\|_{\infty}=0\). What can be said if \(f\) is continuous but not uniformly continuous?

d) Let \(f\) be piecewise continuous. Show that \(\lim _{n ightarrow \infty} f \star \chi_{n}(x)=f(x)\) at all \(x\) where \(f\) is continuous.