Fourier's Theorem/Integral Form

Theorem
Let $f: \R \to \R$ be a real function which satisfies the Dirichlet conditions on $\R$.

Then:
 * $\dfrac {\map f {t^+} + \map f {t^-} } 2 = \displaystyle \int_{-\infty}^\infty e^{2 \pi i t s} \paren {\int_{-\infty}^\infty e^{-2 \pi i d t} \map f t \rd t} \rd s$

where:
 * $\map f {t^+}$ and $\map f {t^-}$ denote the limit from above and the limit from below of $f$ at $t$

if those limits exist.