Continuization is done with a comb of delta function approximations,
multiplied by the discrete function values and summed.
Same as the Riemann sum approximation of a renormalization integral.
The gist of Renormalization is Fuzzyfication:
convolution with the delta function, reproducing the original, is replaced by
convolution with a function that is an approximation to the delta function.
Such a convolution has as a side effect that many singularities are replaced by
indefinite but anyway finite function values.