Overview
Define the delta function as
{% \delta_{\Delta}(t) = \frac{1}{\Delta} %}
for {% 0 < t < \Delta %} and {% 0 %} otherwise.
This is the discrete version of the Dirac Delta Function, which is the delta function in the limit that {% \Delta \rightarrow 0 %}.
Then, a discrete function can be written as
{% x(t) = \sum_{k=-\infty}^{\infty} x(k \Delta) \delta_{\Delta}(t - k\Delta) \Delta %}
Continuous Time
In continuous time, the function can be represented as
{% x(t) = \int_{-\infty}^{\infty} x(\tau)\delta(t-\tau) d \tau %}