A question regarding definition of Deterministic Subexponential Time (SUBEXP)
In the definition of SUBEXP, $\epsilon$ ranges over all positive reals. But you get the same definition if you ask that $\epsilon &lt; \epsilon_0$, for an $\epsilon_0&gt;0$ of your choice; if you ask that $\epsilon$ be rational; if you only go over $\epsilon = 1/n$; and so on. This is because DTIME is monotone: if $f \leq g$ then $\mathsf{DTIME}(f) \subseteq \mathsf{DTIME}(g)$.
An alternative definition of SUBEXP would be:
$$
\mathsf{SUBEXP} = \bigcup_{g(n) = o(1)} \mathsf{DTIME}(2^{n^{g(n)}}),
$$
often denoted simply by $\mathsf{DTIME}(2^{n^{o(1)}})$.
Some examples: $\mathsf{P} \subseteq \mathsf{SUBEXP}$; a function which can be computed in time $2^{n^{1/\log\log n}}$ is in $\mathsf{SUBEXP}$; and a function which can be computed in time $2^{\log^{10} n}$ is in $\mathsf{SUBEXP}$.
In contrast, a function which can be computed in time $2^{n^{1/10}}$ is not necessarily in $\mathsf{SUBEXP}$ (and by the time hierarchy theorem, there is such a function which lies outside $\mathsf{SUBEXP}$).
A function in $\mathsf{DTIME}(2^{n/\log n})$ lies in SUBEPT but not necessarily in SUBEXP.