I'm using Royden & Fitzpatrick's Real Analysis, Fourth Edition, text to prep for an exam. This is one of the questions regarding the general properties of compact metric spaces:
"Let $E$ be a subset of Euclidean space $R^n$. Assume every continuous real-valued function on $E$ is uniformly continuous. Prove that $E$ is closed and bounded."
The errata for this text indicates to replace "and bounded" with "but need not be bounded" - a very different question. In working through, I considered something as simple as $f(x)=x^2$ to be a counterexample to boundedness, and in fact, found some agreement in this post.
Have I made a mistake? Is it safe to assume the errata then contains its own error? Is there another link somewhere that may be more updated that anyone has on hand? (Please let me know if this type of question is inappropriate on this site, thank you.)
Let $n=1$ and $E=\mathbb N$. Then every function from $E$ to $\mathbb R$ is uniformly continuous but $E$ is not bounded. (You can take $f(x)=x^{2}$ as a counter-example, if you wish).