Introduction to Distribution Theory

Motivation and Overview¶

In classical analysis, we work with functions that assign specific values to points in their domain. However, many physical phenomena and mathematical problems require more general objects that can handle singularities, discontinuities, and other irregular behaviors. Distribution theory provides this framework by generalizing the concept of a function.

Distribution (or generalized functions) allow us to:

1. Rigorously define operations like differentiation on functions that are not classically differentiable.

2. Assign meaning to objects like the Dirac delta ``function’’ $\delta(x)$.

3. Provide a unified framework for solving partial differential equations.

4. Handle limiting processes more naturally.