In chapter XII of Hilbert Space, Boundary Value Problems, and Orthogonal Polynomials, By Allan M. Krall.
Gives the example that the solution to $$xf'=0$$ is $$f=c+dH(x)$$ where H(x) is the Heaviside function.
I can't seem to find a proof of this example. Could someone write one here or link me to a complete proof?
Let $T_H$ be the regular distribution corresponding to $H$. Then for $\phi\in C_0^\infty$ we have $$ T_H'\phi = -T_H\phi' = -\int H(x)\phi'(x)\,dx = -\int_0^\infty\phi'(x)\,dx = \phi(0) = \delta\phi. $$ Hence, $T_H' = \delta$, or shorter, $H' = \delta$. From $xf'=0$ you know that $f' = c\delta = (cH)'$. Thus, $f = cH + d$.