Continuous form

see δ-function,
sifting property states

f(τ)=∫−∞∞f(λ)δ(τ−λ)dλf(\tau)=\int_{-\infty}^\infty f(\lambda) \delta (\tau - \lambda) d \lambda

arbitrary function expressed as linear combination of δ\delta functions

Discrete form

expressing an arbitrary discrete function as a linear combination of discrete delta functions:

f(k)=∑i=−∞∞f(i)δ(k−i)f(k) = \sum_{i=-\infty}^\infty f(i) \delta(k-i)