I am reviewing uniform continuity of functions to prepare for the upcoming GRE. I thought it would be useful to have a lot of examples/counterexamples with me. So basically I came up with the following list of true or false, but I am still missing some of the examples/counterexamples.
If $f$ is continuous and $\text{dom}(f)$ is compact, then $f$ is uniformly continuous.
TRUE
If $f$ is continuous and $\text{dom}(f)$ is unbounded, then $f$ cannot be uniformly continuous.
FALSE. $f(x)=x^2$ on $\mathbb{R}$ is not uniformly continuous, but $f(x)=x$ is uniformly continuous.
If $f$ is continuous and $\text{dom}(f)$ is not compact but bounded, then $f$ cannot be uniformly continuous.
Not sure. $f(x)=\frac{1}{x}$ on $(0,1]$ is not uniformly continuous. I remember there's some theorem about "extending" a function but I forgot what it was about.
If $f$ is continuous and has unbounded derivative on $\text{dom}(f)$, then $f$ cannot be uniformly continuous.
I suspect it to be true, but not sure how to show it. First difficulty is that for any $c$, I need to write $|f'(c)|$ in terms of something like $|f(a)-f(b)|$. Mean Value Theorem looks promising, but the direction of implication seems wrong.
The last two are false:
For the second-to-last statement, observe that a constant function is uniformly continuous regardless of the domain.
For the last statement, $f(x)=\sqrt{x}$ is uniformly continuous on $(0,1)$ but has an unbounded derivative on this domain.
I believe the "extension" theorem you're thinking of is the fact that a continuous function on $(a,b)$ is uniformly continuous on this interval if and only if it can be extended to a continuous function on $[a,b]$.