Fourier's Theorem/Integral Form/Continuous Point

Theorem
Let $f: \R \to \R$ be a real function which is Lebesgue integrable. Let $f$ be discontinuous at $t \in \R$.

Then:
 * $\displaystyle \dfrac 1 2 \paren {\map f {t^+} + \map f {t^-} } = \int_{-\infty}^\infty e^{2 \pi i t s} \paren {\int_{-\infty}^\infty e^{-2 \pi i d t} \map f t \rd t} \rd s$

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.