Conditions for finite bandwidth
Let us  stick to traditional functions, considering the almost everywhere equivalence class.  From an actual signal perspective, I am leaving distributions aside.

According to the Paley-Wiener theorem, functions with compact frequency support (corresponding to Bernstein spaces) can be extended
from the real line to the complex plane $\mathbb{C}$ as entire functions
of exponential type; entire functions are holomorphic at all finite points of $\mathbb{C}$ ,  exponential type means that $|f(z)|\le C e^{a|z|}$. They can be called $B$-functions.

So they inherit from all their properties: no bounded support in time, infinite differentiability, Taylor expansion everywhere, controlled decay at infinity...