There is a necessary and sufficient condition for a function $f$ of the real variable to be "band-limited". It is called Wiener-Paley theorem. $f$ must be a restriction on the real line of an entire function of exponential type. Of course, here one implicitly assumes that $f$ belongs to an appropriate space which permits to interpret it as a "signal". For example, $L^2$ (finite energy), or $L^\infty$ (bounded amplitude), or Schwarz temperate distributon etc.

There is a necessary and sufficient condition for a function $f$ of the real variable to be "band-limited". It is called the Wiener-Paley theorem. $f$ must be a restriction on the real line of an entire function of exponential type. Of course, here one implicitly assumes that $f$ belongs to an appropriate space which permits to interpret it as a "signal". For example, $L^2$ (finite energy), or $L^\infty$ (bounded amplitude), or Schwarz temperate distribution etc.

P.S. This is a mathematical website, and the engineering terminology of the problem may sound strange to some mathematicians. So let me try to translate: We are taking about a function of a real variable for which Fourier transform is defined in some sense. Band limited means that Fourier transform has bounded support, and "time-limited" means that the function itself has bounded support. Of course the function cannot be simultaneously time- and band- limited, unless it is zero. This is a (very crude) form of the "indeterminacy principle", and also follows from the Wiener-Paley theorem.