I have a stupid question... if i want to sample a continuous function i can use the Dirac delta distribution
$$f(x_0) = \int_{-\infty}^{+\infty} \delta(x - x_0) f(x)dx,$$
which indeed involves the distribution theory, however i've been thinking that the same thing (with some assumption) could be written as:
$$ f(x_0) = \frac{1}{2 \pi j} \oint_{\gamma} \frac{f(z)}{z - x_0} dz$$
where $\gamma$ is a closed curve that contains $x_0$. Is there some relevant relention/difference between the two representations? Could the Cauchy integral formula be seen as a linear functional (like the one usually studied in functional analysis)?