I want to prove that if $f \in C^{1}(\mathbb{R})$ is compactly supported then its Fourier transform is integrable. I was able to prove the result for $f \in C^{2}(\mathbb{R})$ and compactly supported. I used the fact that if $f \in C^{2}(\mathbb{R})$, then $\hat{f}$ is bounded by $\frac{c}{1+{x}^{2}}$. So it is integrable. I failed to prove it if $f \in C^{1}(\mathbb{R})$

Here is a link to the paper about necessary conditions for the integrability of the Fourier transform: http://www.heldermannverlag.de/gmj/gmj16/gmj16043.pdf It is stated in that paper that sufficient conditions for the integrability of the Fourier transform are given in the book R. M. Trigub and E. S. Bellinsky, Fourier analysis and approximation of functions. Kluwer Academic Publishers, Dordrecht, 2004. 

