I have certain doubts about a classical Fourier inversion theorem. According to it (this is a theorem from "Panorama of Harmonic Analysis" by Krantz), if $f$ and $\hat{f}$ are both in $L_1(R)$ and continuous, then almost everywhere f is equal to the inverse Fourier transform of $\hat f$.

It seems to me that some conditions may be missing here. Indeed, if the function $f$ is the inverse Fourier transform of $\hat f$ then, as a Fourier transform, $f$ must be bounded, uniformly continuous on $R$ and satisfy the conditions $ f(y) \to 0$ as $y \to \infty$ and as $y \to -\infty$. Do these conditions really hold automatically or no if $f$ and $\hat{f}$ are both in $L_1(R)$ and continuous?

Are there any easy-to-check sufficient conditions for a function to be a Fourier transform of a function from $L_1(R)$?

Thanks in advance.