What happens with the Cauchy Integral Formula $\displaystyle \frac{1}{2 \pi i} \int_{\Gamma} \frac{f(\zeta)}{\zeta-z} d \zeta $ as $z$ approaches $\Gamma$? Does the limit exist and in which way should it be considered?

The answer to this question is given by the $L^2(\Gamma)$ boundedness of the Cauchy Integral on Lipschitz curves, that is, the boundedness of the operator:
\begin{equation}
\label{cauchy op}
\tag{CI}
f \mapsto \lim_{\epsilon \to 0} \frac{1}{2\pi i} \int_{\Gamma: |\zeta-z|> \epsilon} \frac{f(\zeta)}{\zeta- z} d \zeta.
\end{equation}

The development of question (\ref{cauchy op}) has been related to understanding another question (AC) in complex analysis:

Let $U$ be an open set in $\mathbb{C}$ and $E \subset U$ a compact subset. Find a "geometric" characterization for the set $E$ with the property that whenever $f: U\setminus E \to \mathbb{C}$ is bounded and analytic, $f$ admits an analytic extension to $U$.

On one hand, (CI) gives a partial, yet satisfactory answer to (AC). On the other hand, progress in (AC) led to generalizations of the classical Calderón-Zygmund theory for singular integral operators and shed some new light in other areas of harmonic analysis.

The purpose of this talk is to point out unexpected associations in mathematics.