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I thought I could use a constant function but I think absolute maximums are attained on a constant function.

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Consider $f:\mathbb{R}\to\mathbb{R}$ with $f(x):=x$. Since for any $x,y\in\mathbb{R}$ we have $|f(x)-f(y)|=|x-y|$, $f$ is uniformly continuous. It does not attain minimum or maximum, as $f$ is unbounded.

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The key here is to choose a domain $E$ that isn't compact. Can you think of a nice, uniformly continuous, function on the open interval $(0,1)$ that doesn't have a maximum or minimum on that interval?

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