Reference for "distributional derivative being zero implies being constant" I know that if a distribution (generalized function) has zero derivative, then it is a constant. I also know the proof. But I have a hard time finding a reference which contains a statement of this fact. Any thoughts? Thanks.
Update:
I indeed found it in "Théorie des distributions" by L. Schwartz. Section 2.4, Theorem 1.
 A: I may be aware that this is very bad timing, but I wanted to point out the fact that there is an easy proof also in $\mathbb{R}^N$. Source: lecture notes by Professor P. D'Ancona (in Italian). https://www1.mat.uniroma1.it/people/dancona/IstAnSup/dispense-esercizi/4-Distribuzioni-20191024.pdf
Proposition. Let $T \in \mathscr{D}^{\prime}\left(\mathbb{R}^{N}\right)$ such that $\nabla T=0 .$ Then there exists a constant $c$ such that $T=T_{c}$, where for $f\in L^1_{loc}$ I denote $T_f$ the distribution represented by $f$.
Proof. Let's define $f_{\varepsilon}:=\rho_{\varepsilon} * T$, the standard regularization of $T$. Then
$$
\nabla f_{\varepsilon}=\nabla\left(\rho_{\varepsilon} * T\right)=\rho_{\varepsilon} *(\nabla T)=0,
$$
thus $f_{\varepsilon}$ is a constant function $f_{\varepsilon}=c_{\varepsilon} .$ On the other hand, $T_{f_{\varepsilon}}=T_{c_{\varepsilon}}$ converge to $T$ in $\mathscr{D}^{\prime}(\Omega)$, hence the sequence of numbers $\left\{c_{\varepsilon}\right\}$ must be convergent, so there exists $c \in \mathbb{R}$ such that $T_{c_{\varepsilon}} \rightarrow T_{c}$.
A: Ok, you said you knew how to prove it. Others may not. And you may like this better than what you have (it's the second thing I always think of when this comes up, and I like it a lot better than the first thing I think of...)
Say $u$ is a distribution and $u'=0$. By definition $u(\phi')=0$ for any test function $\phi$. Hence $u(\phi)=0$ for any test function $\phi$ with $\int\phi=0$.
Fix $\psi_0$ with $\int\psi_0=1$, and let $c=u(\psi_0)$. Now for an arbitrary test function $\phi$, let $\alpha=\int\phi$. Then $u(\alpha\psi_0-\phi)=0$, which says $$u(\phi)=c\int\phi.$$Which is exactly what "$u=c$" means.

Detail: We used the following fact above: Given a test function $\phi$ on $\Bbb R$, there exists a test function $\psi$ with $\phi=\psi'$ if and only if $\int\phi=0$. In case this is not clear: First, if $\phi=\psi'$ then $\int\phi=\int\psi'=0$ because $\psi$ has compact support. Suppose on the other hand that $\int\phi=0$, and define $\psi(x)=\int_{-\infty}^x\phi$. Then $\psi'=\phi$ and hence $\psi$ is infinitely differentiable, while the fact that $\int\phi=0$ shows that $\psi$ has compact support.
