Separating $DSPACE(f)$ and $DTIME(2^f)$: is there any function computable in time $2^{O(f)}$ but not in space $O(f)$? For $f(n)\ge n$, 

$$\mathsf{DSPACE}(f(n)) \subseteq \mathsf{DTIME}(2^{O(f(n))}).$$ 

Is there any function $f$ for which this containment is known to be proper?
If $\mathsf{L} = \mathsf{P}$ then $\mathsf{DSPACE}(O(f(n))) = \mathsf{DTIME}(2^{O(f(n)})$ for all time-constructible functions $f(n) \geq \log n$ using standard padding arguments. I am not sure what happens if $f(n)$ is not time-constructible.

If $\mathsf{L} \neq \mathsf{P}$ then $f(n) = \log n$ will be an example. Therefore there is a time-constructable function $f(n)$ where the containment is proper iff the containment is proper for $f(n) = \log n$ .