Example of a function f and a set E with the following: f is uniformly continuous on E, but f doesn't attain either a max or a min on E.

Question

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

Answer 1

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?

Answer 2

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.