In my work, I am encountering the issue of having to multiply a continuous function (not necessarily differentiable) by a distribution.
It seems to me that if $f(x)$ is a continuous function on $\mathbb{R}$ and if $d(x)$ is a distribution on $\mathbb{R}$, then the product $f(x) d(x)$ makes sense and can be interpreted as a distribution.
If someone knows an example of a distribution, which cannot be multiplied by a continuous function, I would like to see it. If there is no such example, then my question is this: is there a reference discussing the extent with which distributions can be multiplied together? Hopefully, in such a discussion, a particular case would be the case permitting the multiplication of a distribution by a continuous function. My goal being to justify my operations with theory.
If you take for example the Dirac delta distribution at $x=0$ and a continuous function $f(x)=\max(0,x)$, you find very quickly that the properties of Schwarz distributions break down. The product would have to be defined as the functional $\phi\mapsto f(0)\phi(0)$ for all $\phi\in C_0^\infty(\mathbb{R})$, which would give you zero. When you try to differentiate this distribution $\delta\cdot f$, you will not be able to. The only reason that you can have very non-differentiable distributions is that the test function space is extremely nice. The smoother the test function space is, the more horrible the distribution space can be. If you want to be able to multiply by merely continuous functions, you could limit your distributions to Radon measures, for example. Then your distribution space will be the dual of the continuous functions with compact support. See wikipedia