For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
Show that $T_{\mu}$ is a linear map, which sends the space $\mathcal{C}_{c}(\mathbb{R})$ into itself, and that for all $\varphi \in \mathcal{C}_{c}(\mathbb{R})$ we have $\|T_{\mu}\varphi\|_{\infty} \leqslant \|\varphi\|_{\infty}$.