In trying to answer this question, I thought that it might be useful if there exists a function $K:\mathbb R^2\to\mathbb R$ such that for any continuous function $f:\mathbb R\to\mathbb R$, $$\int_{-\infty}^\infty f(x)K(f(x),x)dx<\infty.$$ At first I thought about Schwartz-class functions, like $K(f(x),x)=e^{-x^2}$, but this would not work for functions with exponential growth. In fact, I realized, that $f(x)$ could grow arbitrarily fast, $K$ would have to be a function of $f(x)$. I though maybe $e^{-|f(x)|}$ might work, but this fails miserably for $f(x)=\ln(x)$. Now I'm thinking something like $e^{-f^2(x)-x^2}$, but I don't know how to prove integrability.
It seems reasonable that such a function exists, but on the other hand, one construct that would have this property is the so-called Dirac-delta function. This is not truly a function so I wonder if another construct had this property, it would also not be a function.
Does such a function $K$ exist? If so, how are its properties proved?
Edit:
Thanks to the clever answer provided by @CorandoCosta, I should clarify that I am wondering about such a function $K$ with the stipulation that $f(x)K(f(x),x)\neq g(x)K(g(x),x)$ for $f(x)\neq g(x)$. Also, $K(f(x),x)$ should not have compact support, in general. Maybe even restrict that $K$ is nonzero except for on a null set.
Yes, there is such a function, take $K(y,x) = \frac{1}{y}\frac{1}{1+x^2+y^2}$ when $y \neq 0$ and $K(0,x)= 0$
Note that $$\int f(x) K(f(x), x)\, dx = \int \frac{1}{1+x^2+f(x)^2} \, dx < \infty$$