How is the space hierarchy theorem different for non space constructible functions?
Your question is answered on Wikipedia: for any $s(n)$, and $f(n) = O(s(n))$, and any computable $g(n) = o(s(n))$, $\mathsf{DSPACE}(f(n)) \neq \mathsf{DSPACE}(g(n))$. This is a result due to Geffert.