In here it was clarified that $\cap_{f(n)\in\omega(1)}\mathcal C(n^{f(n)})\subsetneq\cap_{\epsilon>0}\mathcal C(n^{n^\epsilon})$ where $\mathcal C(t(n))$ is the class of problems solvable in $O(t(n))$ time. My query now is whether there is a function of $g(n,\epsilon)$ that non-trivially depends on $\epsilon$ with $g(n,\epsilon)<g(n',\epsilon')\implies\epsilon<\epsilon'\vee n<n'$ such that $\cap_{f(n)\in\omega(1)}\mathcal C(n^{f(n)})=\cap_{\substack{\epsilon>0}}\mathcal C(n^{g(n,\epsilon)})$ can hold?That is I seek a different characterization of $\mathsf{P}$.
On different characterizations of $\mathsf P$
complexity theory;complexity classes;polynomial time