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 $p=\infty$.
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Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please [edit] the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – José Carlos Santos Nov 21 '19 at 08:45
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Where is the question – Cloud JR K Nov 21 '19 at 09:26
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I upvoted because of the first question. Welcome to MSE – Cloud JR K Nov 21 '19 at 09:28
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Related: https://math.stackexchange.com/q/69687/169085 – Alp Uzman Aug 01 '22 at 16:51
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$\|I_{(a,b)}(x)-I_{(a,b)}(x-h))\|_{\infty} =1$ whenever $h \neq 0$.
Kavi Rama Murthy
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It is the function which has the value $1$ on the interval $(a,b)$ and $0$ outside the interval. – Kavi Rama Murthy Nov 21 '19 at 08:49
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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).
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You have to consider a bounded continuous function which is not uniformly continuous. – Kavi Rama Murthy Nov 21 '19 at 08:54
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