Generalized functions, also known as distributions, extend the concept of functions to include objects that may not be well-defined in the traditional sense. They allow for the manipulation of functions that can represent physical phenomena, such as Dirac delta functions, which are useful in physics and engineering for modeling point sources or impulses. These functions are defined through their action on test functions, which are smooth and well-behaved. This framework enables mathematicians and scientists to work with singularities and discontinuities, providing a powerful tool for solving differential equations and analyzing signal processing and quantum mechanics.