Let's consider the recursive function defined as follows:
$$T(n)=T(\frac{n}{2})+1$$
However, it's important to clarify the domain of this function, specifically, the values of '$n$' for which it is defined. Is this function only defined for '$n$' that are powers of $2$, or can we also define it for non-power-of-$2$ values, such as '$T(3)$'?
Also what about this one?
$$T(n)=T(\frac{n}{2})+T(\frac{n}{3})+1$$
Good question. For the problems we encounter in the real world in computer science, it usually doesn't matter to the asymptotic running time, so we typically gloss over these details, to avoid bogging down the reader in irrelevant trivia. You can consider the domain to be all integers and insert some floors/ceilings, or consider the domain to be powers of two. Since running times are normally monotonically increasing, once you prove an asymptotic result about $T(n)$ for powers of two, it typically holds for all integers, too.
All of this is arguably a bit sloppy, but once you gain some experience hopefully it will seem more reasonable.
If you want to look into the specific details, see https://cs.stackexchange.com/q/9569/755, https://cs.stackexchange.com/q/152809/755, https://cs.stackexchange.com/q/61260/755, https://cs.stackexchange.com/q/74431/755, https://cs.stackexchange.com/q/52298/755.