Problems solvable in time $f(n)$ but not in time $o(f(n))$
Consider the language of all binary words which consist only of zeroes. It can be decided on a deterministic Turing machine in time $O(n)$, but not in time $o(n)$.
For general time-constructible functions $f(n)$, the time hierarchy theorem shows that the problem of deciding whether a Turing machine halts in time $f(n)$ can be solved in time $O(f(n)\log f(n))$ but not in time $o(f(n))$. It is not known how to eliminate this logarithmic gap in general.