An Example of a Theorem that has Contradictory Relativizations and a Diagonalization Proof

We construct a computable space bound $S(n)$, with $n^{2} less than S(n) less than n^{3}$ and show by diagonalization that DSPACE [$S(n)$] = DSPACE [$S(n)$ log $n$]. Moreover, we can show that there exists an oracle $A$ such that DSPACE [$S(n)$] $\neq$ DSPACE$^{A}$[$S(n)$ log $n$]. This is a counterexample to the belief that is a theorem has contradictory relativizations, then is cannot be proved using standard techniques like diagonalization [7].