The Gaps

Note that there is a polynomial gap on either side. It's not enough for f(n) to be bigger (or smaller) than $n^{\log_b(a)}$, it has to be polynomially so (i.e., the $\epsilon\gt$ matters).

In one case, we can partially fill this gap. That is, if f(n) is just a log factor bigger than $n^{\log_b(a)}$, we still get a contribution from each level: $T(n) = \Theta(n^{\log_b(a)} \log^{k+1} n)$.