# Why don't we multiply generalized functions?

• A
Because it drives to contradictions. Here is a nice example from E. Rosinger Generalized solutions of nonlinear PDE.

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

$$\int f(x) x \delta(x) dx= 0$$