Misconception about step function and Heaviside distribution

Question

Given the step function: ${\displaystyle h(x):={\begin{cases}1,&x\geq0\\0,&x<0\end{cases}}}$

How is the Heaviside distribution $H \in {\cal D'}(\mathbb R)$ defined?

1. $ H (f) = h(x) \qquad \forall f \in {\cal D}(\mathbb R) $

or

2. $ H (f) = {\displaystyle \int_\mathbb R} h(x) f(x) dx \qquad \forall f \in {\cal D}(\mathbb R)$

Answer 1

The Heaviside distribution is defined by $$ H(f) = \int_\mathbb R h(x) f(x) \, dx = \int_0^\infty f(x) \, dx. $$

But when a distribution is given by a function, $T(f) = \int t(x) \, f(x) \, dx,$ the function $t$ and the distribution $T$ are often identified. So it is common to just define $t$ and refer to that as a distribution.