For every computable function $f$ does there exist a problem that can be solved at best in $\Theta(f(n))$ time?
For every computable function $f$ does there exist a problem that can be solved at best in $\Theta(f(n))$ time or is there a computable function $f$ such that every problem that can be solved in $O(f(n))$ can also be solved in $o(f(n))$ time?


If $f$ is a time-constructible function, then the Time Hierarchy Theorem says that there are problems that require $O(f(n))$ time, and cannot be solved with time $o\left (\frac{f(n)}{\log(f(n))}\right )$. Specifically,
$$ \text{DTIME}\left ( o\left (\frac{f(n)}{\log(f(n))}\right) \right ) \subsetneq \text{DTIME} \left ( f(n) \right)$$

This consider only decision problems, and does not regard the time it takes to generate the output.