Is DTIME(n) = DTIME(2n) true? (unlike Rosenberg's results)
$\mathrm{DTIME}(O(n))$ is the set of problems that can be solved in deterministic $O(n)$ time for some constant implicit in $O$, in other words, it is the union of the $\mathrm{DTIME}(cn)$ for all $c&gt;0$.  That this union is, in fact, equal to $\mathrm{DTIME}(cn)$ for any given $c&gt;1$ (i.e., $1+\varepsilon$) means that all the $\mathrm{DTIME}(cn)$ for $c&gt;1$ are equal, it does not say anything about $\mathrm{DTIME}(n)$, so there is no contradiction with the fact that $\mathrm{DTIME}(n)\neq \mathrm{DTIME}(2n)$ (although the results could have been stated in a clearer way).