Special case: Band-limited in time domain and band-limited in frequency domain
The condition is that the function $f(t)=0$, if you are talking about continuous time signals and frequencies (some of your formulations are confusing to me). In other words, a (nonzero) time-limited signal cannot be also band-limited. In other words, a function and its (continuous) Fourier transform cannot both have finite support. 

The core of one proof (using complex analysis) relies on Paley-Wiener theorems: the Fourier transform of a time-limited function would be an entire function (a complex-valued function, holomorphic over the whole complex plane), that would vanish on an open interval, and hence be zero (almost) everywhere.

You can find details in Fourier transforms of compactly supported functions, and the paper (for instance) The uncertainty principle for Fourier transforms on the real line. This is a reason for the development of time-frequency analysis.