Must an operator $\mathcal{O}$ satisfying $\mathcal{O} (f + c) = \mathcal{O} (f)$ for any constant function $c$ be a differential operator?

Question

Consider an operator (not necessarily linear) $\mathcal{O}: C^{\infty} (\mathbb{R}) \to C^\infty(\mathbb{R})$ which satisfies the following property for any function $f \in C^\infty(\mathbb{R})$ and any constant function $c \in C^\infty(\mathbb{R})$ \begin{equation} \mathcal{O} \left(f + c \right) = \mathcal{O} \left(f \right) \end{equation}

Must the operator $\mathcal{O}$ be a differential operator? If not, what additional conditions should $\mathcal{O}$ have for it to have to be a differential operator? For example, would $\mathcal{O}$ being a linear operator suffice?

Answer 1

No; there are many many linear operators which satisfy this condition. For example, let $\mathcal{O}$ be the operator which sends a smooth function $f$ to the constant function with value $f(1)-f(0)$.

See here for a characterization of the derivative, which then leads to a characterization of differential operators.