This is embarrassing.
I asked this question several months ago:
Let $f$, a Lebesgue integrable function in $\mathbb{R}$ ($\int_{\mathbb{R}}|f| < \infty$). Let: $g(t) := \int_{-\infty}^{\infty} f(x)\sin(tx)dx.$ Show that $g$ is continuous and that: $ \lim_{|t| \rightarrow \infty} g(t) = 0. $
I wrote there that I proved that $g$ is continuous, but I can't for life of me remember my proof, come up with a new one or find my notes. Help?