Singularity functions (also called switching functions) are very useful in circuit analysis. They serve as good approximations to the switching signals that arise in circuits with switching operations.
Singularity functions are functions that either are discontinuous or have discontinuous derivatives.
The three most widely used singularity functions in circuit analysis are the unit step, the unit impulse, and the unit ramp functions.
The three singularity functions are related by differentiation as
\(\displaystyle \delta\left(t\right) = \frac{d}{dt} u\left(t\right)\)
\(\displaystyle u\left(t\right) = \frac{d}{dt} r\left(t\right)\)
or by integration as
\(\displaystyle r\left(t\right) = \int_{-\infty}^{t} u\left(\lambda\right)\,d\lambda\)
\(\displaystyle u\left(t\right) = \int_{-\infty}^{t} \delta\left(\lambda\right)\,d\lambda\)