An explicit separation between time-constructibility and space-constructibility? Show a function $f(n)$ which is space-constructible but not time-constuctible. 

Is this problem related to a possible separation between complexity classes DTIME(f(n)) and SPACE(f(n))?
A function $T \colon \mathbb{N} \to \mathbb{N}$ is time constructible if there is a Turing machine $M$ which, on input $1^n$, computes the function $x \mapsto T(|x|)$ in time $O(T(n))$.

A function $S \colon \mathbb{N} \to \mathbb{N}$ is space constructible if there is a Turing machine $M$ which, on input $1^n$, computes the function $x \mapsto S(|x|)$ in space $O(S(n))$.

Some texts require that time/space constructible functions be non-decreasing. Some texts require the time constructible functions satisfy $T(n)\ge n$, and the space constructible functions satisfy $S(n)\ge \log n$. Some texts do not make use of the $O(\cdot)$ notation in the definition.

Anyway, it is easy to show that every "ordinary" function $f$, satisfying $f(n)\ge \log n$ and $f(n) = o(n)$ is space constructible, but not time constructible.

The constructibility problem is not directly related to possible separation between complexity classes DTIME(f(n)) and SPACE(f(n)). However, the statement of time and space hierarchy theorems incorporates the constructibility. For example:

Time Hierarchy Theorem
If $f$, $g$ are time-constructible functions satisfying $f(n)\log f(n) = o(g(n))$, then
$\mathbf{DTIME}(f(n))$ is a proper subset of $\mathbf{DTIME}(g(n))$.

See Arora &amp; Barak's book or Papadimitriou's for more information. (The latter uses the term "proper complexity function" to refer to one that is both time and space constructible.)