Suppose $f(t)$ is a function whose Fourier transform $\hat{f}(\omega) = \int_{-\infty}^{+\infty} f(t) e^{- \omega t} dt$ is supported on the interval $[-\epsilon,+\epsilon]$. Is there a theorem to the effect that $f$ can't change too fast? Intuitively, it seems like this ought to be the case since $f$ is ``composed'' only out of slowly varying sines and cosines.
For example, will it be true that if we restrict $||f||=1$ for some appropriate norm, then some bound on $\sup_t |f'(t)|$ will hold? If not, are there other ways to make the preceeding paragraph precise?
P.S. This is related to a different question I asked a while ago.