- The step distribution, $step(f) \equiv \int_0^\infty f(x) dx$
- The dirac delta distribution, $\delta(f) = f(0)$.
- Notationally, we write $D(f)$ where $D$ is a distribution, $f$ is a function as $\int D(x) f(x) dx$.
- We can regard any function as a distribution, by sending $f$ to $F(g) \equiv \int f(x) g(x) dx$.
- But it also lets us cook up "functions" like the dirac delta which cannot actually exist as a function. So we move to the wider world of distributions

$\begin{aligned}
&\int_0^\infty U dV = [UV]|_0^\infty - \int_0^\infty V dU \\
&\int_0^\infty f(x) D'(x) dx \\
&= [f(x) D(x)]|_0^\infty - \int_0^\infty D f'(x)
\end{aligned}$

Here, we assume that $f(\infty) = 0$, ahd that $f$ is differentiable at $0$.
This is because we only allow ourselves to feed into these distribtions
certain classes of functions (test functions), which are "nice". The test
functions $f$ (a) decay at infinity, and (b) are smooth.
The derivation is:
$\begin{aligned}
&\int_0^\infty U dV = \int_0^\infty f(x) \delta(x) = f(0) \\
&[UV]|_0^\infty - \int_0^\infty V dU = [f(x) step(x)]|_0^\infty - \int_0^\infty step(x) f'(x) \\
&= [f(\infty)step(\infty) - f(0)step(0)] - step(f') \\
&= [0 - 0] - (\int_0^\infty f'(x) dx) \\
&= 0 - (f(\infty) - f(0)) \\
&= 0 - (0 - f(0)) \\
&= f(0)
\end{aligned}$

- Thus, the derivative of the step distribution is the dirac delta distribution.