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).

EDIT. In light of the answers given here, it is apparent that I have not defined the function space clearly. Unfortunately, I do not know of a definition for these so I can only state the properties which I may assume, and which seem relavent.

Firstly, $F$ is analytic and not $L^1$. $L^p$ results for larger $p$ are definitely of interest to me.

Secondly, by "everywhere defined", I mean that $f$ has no spikes, but continuity is too strong- I want to allow $f$ to have jumps for arbitrarily large values of $x$.

Further, the improper integral should be understood as $\lim_{R\rightarrow\infty}\int_{-R}^{R}$.