- #1

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We can multiply generalized functions from ##\mathcal D'(\mathbb{R})## by functions from ##C^\infty(\mathbb{R})##. This operation is well defined. For example $$x\delta(x)=0\in \mathcal D'(\mathbb{R}),\quad x\cdot\frac{1}{x}=1\in \mathcal D'(\mathbb{R}),\quad \frac{1}{x}\in \mathcal D'(\mathbb{R}).$$

On the other hand ##C^\infty(\mathbb{R})\subset \mathcal D'(\mathbb{R})##

Ok then:)

$$\delta=\Big(x\cdot\frac{1}{x}\Big)\cdot\delta=\frac{1}{x}\cdot(x\delta)=0.$$