Is $\mathsf{DTIME}(n) = \mathsf{DTIME}(2n)$? Define $\mathsf{DTIME}(f(n))$ as the class of languages that can be accepted by a (multitape) Turing machine in time $f(n) + 1$. (The "$+ 1$" is just to simplify notation and avoid confusion.) Notice that there is no $O(\cdot)$ around $f(n) + 1$.

Is it true that $\mathsf{DTIME}(n) = \mathsf{DTIME}(2n)$?

Using the linear speed-up theorem, we can prove $\mathsf{DTIME}(2n) = \mathsf{DTIME}(1.01n)$, but can we reach $n$?

It seems that the language of palindromes is in $\mathsf{DTIME}(n)$; for related topics, see Lipton's blog post about string algorithms
From the comment: 

In "Deterministic Turing Machines in the Range between Real-Time and Linear-Time" I found:

... if $r \in T^{−1}(DTM)$ and $r' \in o(r)$ then $DTIME(n+r') \subset DTIME(n+r)$ ...