A funtion and its fourier transformation cannot both be compactly supported unless f=0 Problem : Suppose that $f$ is continuous on $\mathbb{R}$. Show that $f$ and $\hat f$ cannot both be compactly supported unless $f=0$.
Hint : Assume $f$ is supported in [0,1/2]. Expand $f$ in a Fourier series in the interval [-,1], and note that as a result, f is a trigonometric polynomial.
I proved that f is trigonometric polynomial by using hint.
But, I don't know how to prove function's fourier transform cannot compactly supported function. 
Can I get some hints?

By the Schwartz's Paley-Wiener theorem the Fourier transform of a compact-supported function is an entire function, and non-zero entire functions cannot be compact supported by Liouville's theorem.
