"The running time of an algorithm $t(n)$ is $(g(n))$ if and only if its worst$-$case running time $t_w(n)$ is $O(g(n))$ and its best$-$case running time $t_b(n)$ is $(g(n){)}^{\text{'}\text{'}}$

Notes:

There is no mention of the average case running time here:

$t(n)$ is a function representing the running time of an algorithm in general, not for a specific input class. It may have a different closed form for each input class, or no closed form at all.

$t_w(n)$ is a function representing the running time of an algorithm for the specific class of inputs designated as "worst case".

$t_b(n)$ is a function representing the running time of an algorithm for the specific class of inputs designated as "best case".

This is an $"$if and only if$"$ statement, so the proof usually involves two parts.