Asymptotic growth rate of $f(n)$ and $f(n+1)$
Imagine any continuous, monotonically non-decreasing function $f$ such that $f(n) = n!$ for all non-negative integers $n$. Then $f(n+1) = (n+1)! = (n+1)n! = (n+1)f(n)$. Since there is no constant $c$ such that $n+1&lt;c$, it follows that it is not true that $f(n+1) = \Theta(f(n))$. QED.