Prove that $o(g(n)) \cap \omega(g(n))$ is the empty set.

By definition, $o(g(n))$ is the set of functions $f(n)$ such that $% $ for any positive constant $c_1 > 0$ and all $n \ge n_0$. $\omega(g(n))$ is the set of functions $f(n)$ such that $% $ for any positive constant $c_2 > 0$ and all $n \ge n_0$.

So, $o(g(n)) \cap \omega(g(n))$ is the set of functions $f(n)$ such that $% $. Now, this inequality cannot be true asymptotically as $n$ becomes very large, $f(n)$ cannot be simultaneously greater than $c_2g(n)$ and less than $c_1g(n)$ for any constants $c_1, c_2 > 0$. Hence, no such $f(n)$ exists.

Another way to look at this is By definition,

and

Both of this cannot be simultaneously true. Hence, no such $f(n)$ exists.