Give a counter-example to show translation is not continuous on $L^p$ when $p = \infty$

Question

A function $f:\mathbb{R}^n\to\mathbb{C}$ is said to be $L^p$-continuous if $\tau_h(f)\to f$ in $L^p(\mathbb{R}^n;\mathbb{C})$ as $h\to 0$ in $\mathbb{R}^n$, where $\tau_h(f)(x)=f(x-h)$ is the translation of $f$ by $h$. If $1\leq p< \infty$ every $f\in L^p(\mathbb{R}^n;\mathbb{C})$ is $L^p$-continuous. Give a counter-example to show this result is not true when $p=\infty$.

Answer 1

$\|I_{(a,b)}(x)-I_{(a,b)}(x-h))\|_{\infty} =1$ whenever $h \neq 0$.

Answer 2

For instance, consider any function $f\in L^\infty(\Bbb R^n)$ which is continuous but not uniformly continuous (exercise for you is to see where continuity comes into play).