I would like something like this to be true:
Conjecture: There is a function $g(n)$ such that for all functions $f(n)$ (perhaps satisfying some reasonable properties, like time-constructability), there is a language in $TIME[f(n)]$ that is not in $TIME[f(n) - g(n)]$.
Is anything like this known? Can this be proven under any reasonable assumptions? Would this have any interesting consequences?
By $TIME[f(n)]$, I mean the set of problems solved by a Turing machine that halts in exactly $f(n)$ steps or less - not to be confused with $O(f(n))$ or less, in which case the conjecture is trivially false when $f(n) >> g(n)$.