I am looking for a theorem that guarantees the polynomial growth of a function $f$ defined by a Fourier integral, that is, when $$f(x)=\int_{-\infty}^{\infty}F(y)e^{ixy}dy.$$ I am only interested in one-sided growth, say $x\rightarrow +\infty$. Further, in the case I have in mind, $f$ is almost everywhere continuous, and everywhere defined.

It seems to me that all such functions should be of polynomial growth, but I can't see how to prove it (beyond the cases when the integral is absolutely convergent, which is too strong for my purposes).