ABSTRACT
A generalized function may be defined multiplying by an ordinary “test” function and integrating over the real line, provided this leads to a finite unique result. For example, taking as test functions the functions integrable on the real axis, multiplication by the unit jump limits the integral to the positive real axis (Subsection 3.1.1); thus the unit jump is defined as the functional that assigns to each test function integrable on the real axis a number, namely, its integral on the positive real axis. This leads to the fundamental property, the unit jump (Subsection 3.1.2). The unit impulse is defined as the derivative of the unit jump; multiplying by the test function, and integrating by parts, requires the test function to be differentiable on the real axis. This leads to the definition of the unit impulse as the functional that assigns to each differentiable test function on the real line a number, namely, its value at the origin (Subsection 3.1.3); this is the fundamental property of the unit impulse that can be extended to continuous functions and leads to linear transformation (Subsection 3.1.4) including change of scale and translation.
