# Dirac delta function divided by Dirac delta function

Is the following defined: (Dirac delta function divided by Dirac delta function)

$$f = \frac{\delta}{\delta} = ?$$

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I don't know if the following is what you are looking for, but: To give the division a sense, what you can do is look for functions $\phi \in \mathcal E^0(\mathbb R)$ (that is continuous functions $\mathbb R \to \mathbb R$ that fulfill $\phi \delta = \delta$. As for any $\psi \in \mathcal D(\mathbb R)$ we have $$(\phi \delta)(\psi) =\delta(\phi\psi) = \phi(0)\psi(0) = \phi(0)\delta(\psi),$$ that is $\phi \delta = \phi(0)\delta$, so we have $$\phi \delta = \delta \iff \phi(0) = 1.$$
Remember $\delta$ is 'operationally' defined by its effect under integration. So $\delta$/$\delta$ needs more context, for example, like
($\delta$(x-y),f(x))/($\delta$(x-y),g(x)),
where (.,.) is the inner product, before the question can be answered. In this case $\delta$/$\delta$ = 1 only if f(y)=g(y). Otherwise the results could be one of many other things.