Extension of the Dirac delta function

Question

For real number $a \in \mathbb{R}$, the heaviside step function $H_{a} : \mathbb{R} \to \{0,1\}$ is usually defined as \begin{equation} H_a(x) = \begin{cases} 1, & x \geq a; \\ 0, & x < a. \end{cases} \end{equation} From wikipedia I learned that using distribution theory, the Dirac delta function $\delta_a$ can be formulated as a distributional derivative of the heaviside step function $H_a$. My question is: if we extend the definition of the heaviside step functions to include the case $a = \infty$, i.e., we define an extended real-valued function $H_{\infty}: \mathbb{R} \cup \{\infty\} \to \{0,1\}$ as \begin{equation} H_{\infty}(x) = \begin{cases} 1, & x = \infty; \\ 0, & x \in \mathbb{R}, \end{cases} \end{equation} then is it still possible to extend the Dirac delta function to an extended one $\delta_{\infty}$ as a distributional derivative with respect to $H_{\infty}$? Anyone has an idea? Thanks very much.