If all space-constructible functions are time-constructible, then $EXP-TIME=EXP-SPACE$. To prove that (and to give an example of a non-trivial space-constructible but presumably not time-constructible function), let us take an arbitrary (possibly $EXP-SPACE-COMPLETE$) problem $L\in EXP-SPACE$, $L\subseteq\{0,1\}^*$. Then there exists a $k\in\mathbb{N}$, s.t. $L$ can be solved by a DTM $M$ in $2^{n^k}$ space. Now define the function $$f(n)=\left\{\begin{array}{ll} 8n+2 & \mbox{if }\left(\mbox{first } \lfloor\sqrt[k]{\lfloor\log n\rfloor+1}\rfloor\mbox{ bits of } bin(n)\right)\in L\\ 8n+1 & \mbox{else} \end{array} \right.$$

The condition can be decided in $2n$ space, thus $f$ is space-constructible. If $f$ was time constructible, then it is easy to see that we could solve $L$ in exponential time.

This answer uses the same idea.

If all space-constructible functions are time-constructible, then $EXP-TIME=EXP-SPACE$. To prove that (and to give an example of a non-trivial space-constructible but presumably not time-constructible function), let us take an arbitrary (possibly $EXP-SPACE-COMPLETE$) problem $L\in EXP-SPACE$, $L\subseteq\{0,1\}^*$. Then there exists a $k\in\mathbb{N}$, s.t. $L$ can be solved by a DTM $M$ in $2^{n^k}$ space. Now define the function $$f(n)=\left\{\begin{array}{ll} 8n+2 & \mbox{if }\left(\mbox{first } \lfloor\sqrt[k]{\lfloor\log n\rfloor+1}\rfloor\mbox{ bits of } bin(n)\right)\in L\\ 8n+1 & \mbox{else} \end{array} \right.$$

The condition can be decided in $2n$ space, thus $f$ is space-constructible. If $f$ was time constructible, then it is easy to see that we could solve $L$ in exponential time.

This answer uses the same idea.