Let $ f\in L^1(\mathbb{R}) $. Show that $ \hat{f}(x) $ is continuous and $ \lim_{|x|\to\infty} \hat{f}(x)=0 $.
Proof: To show continuity, we only need to show that if $ x_k\to x $ then $ \hat{f}(x_k)\to\hat{f}(x) $
$ \lim_{k\to\infty}\hat{f}(x_k)=\lim_{k\to\infty}\int e^{-ix_kt}f(t)dt = \int e^{-ixt}f(t)dt = \hat{f}(x) $
We can pass this limit through the integral since $ \hat{f} $ is dominated by $ f\in L^1 $
