Regularity condition in the master Theorem in the presence of Landau notation for f
Suppose that $T(n) = aT(n/b) + \Theta(f(n))$. What this means is that $T(n) = aT(n/b) + g(n)$, where $g(n) = \Theta(f(n))$, say $c_{LB} f(n) \leq g(n) \leq c_{UB} f(n)$. Consider now the following two recurrences:
$$
T_{LB}(n) = aT_{LB}(n/b) + c_{LB} f(n), \\
T_{UB}(n) = aT_{UB}(n/b) + c_{UB} f(n),
$$
with the same initial conditions as the initial recurrence. You can show inductively that $T_{LB}(n) \leq T(n) \leq T_{UB}(n)$. Assuming that the function $f(n)$ does satisfy the regularity conditions, you can use the master theorem to solve the two auxiliary recurrences, and hence to deduce the solution to the original recurrence.