Fourier inversion of $f(k) \Leftrightarrow \lim_{k \to -\infty } f(k)=0$

Question

Let $k \to f(k)$ be a function and define its Fourier transform as $$ \hat{f} (u) = \int_{-\infty}^{\infty} e^{iux} f(x) dx $$ if $\hat{f} (u)$ is integrable we can get back $f$ by doing the inversion $$ f(k)=\frac{1}{2\pi} \int_{-\infty}^{\infty} e^{-iux} \hat{f}(u) du $$ now a source I am reading indicates (if I understand it correctly) that this can be done if and only if $$ \lim_{k\to -\infty} f(k) =0 $$ is this true? If so how come?