A problem about asymptotic functions
Let $f(n) = \begin{cases} 
1 &amp; \mbox{if $n$ is odd}\\
n &amp; \mbox{if $n$ even}
\end{cases}$, and $g(n) = \begin{cases} 
n &amp; \mbox{if $n$ is odd}\\
1 &amp; \mbox{if $n$ even}
\end{cases}$.
Then, $h(n)=f(n)+g(n) = n+1 \not\in O(f(n))$.
Indeed, for any $n_0$ and $c&gt;0$, there is some $n \ge n_0$ such that $h(n) &gt; c f(n)$.
Simply pick $n$ as an odd integer greater than $\max\{c, n_0\}$, e.g., $n=2\max\{c, n_0\}+1$. Then $h(n) = n+1 = 2\max\{c, n_0\} + 2 &gt; c = c \cdot 1 = c f(n)$.
A similar argument shows that $h(n) \not\in O(g(n))$.