  Let $f$ be a function on the real line. The Fourier transform inversion
theorem is proved under the assumption that $f$ is absolutely continuous such
that $f$ and $f'$ are Lebesgue integrable. A function $g$ is defined by
$f'(t)-iwf(t)=g(t)$. This differential equation has a well known integral
solution using the Heaviside step function. An elementary calculation with
residues is used to write the Heaviside step function as a simple contour
integral. The rest of the proof requires elementary manipulation of integrals.
Hence, the Fourier transform inversion theorem is proved with very little
machinery. With only minor changes the method is also used to prove the
inversion theorem for functions of several variables and to prove Riemann's
localization theorem.
