Showing $\mathbb{R}^p\ni t \mapsto f_t \in L_\infty(\mathbb{R}^p, \mathcal B_p, \lambda_p, \mathbb{C})$ is not continuous

Question

Let $f_t(x):=f(x+t)$

I want to show that $f \mapsto f_t$ is a linear, isometric bijection from $L_\infty(\mathbb{R}^p, \mathcal B_p, \lambda_p, \mathbb{C})\to L_\infty(\mathbb{R}^p, \mathcal B_p, \lambda_p, \mathbb{C})$

$f \mapsto f_t$ is obviously linear and bijective, the inverse is $f \mapsto f_{-t}$. Because of the translation-invariance of $\lambda_p$ we have $\|f_t\|_p=\|f\|_p$ and therefore the mapping is a linear, bijective isomorphism.

How can I show that for arbitrary $f \in L_\infty(\mathbb{R}^p, \mathcal B_p, \lambda_p, \mathbb{C})$ the mapping $\mathbb{R}^p\ni t \mapsto f_t \in L_\infty(\mathbb{R}^p, \mathcal B_p, \lambda_p, \mathbb{C})$ is not continuous?

Answer 1

Consider $p=1$ and $f$ the indicator of the unit interval. Then $\left\lVert f_t-f_s\right\rVert_\infty=1$ for all distinct real numbers $s$ and $t$.