Can we construct an unbounded derivation on abelian C* algebra which is not closable?

One of possible construction may be found in the paper by Bratteli and Robinson(Unbounded derivations of C*-algebras). In the paper they construct $\delta_{0}$ in theorem 15. The only question is that there are no "theorem 15" in this paper. I doubt this is a typo. The only related theorem seems to be theorem 12, but $\delta_{0}$ there is differentiation, which is closable. So I don't know if that counts.

Edit In the comments @Narutaka_OZAMA provide an example of non-closed derivation by setting $D(t)=1$, $D(e^{t})=0$. However I fail to see why this example is well defined. If $D(t)$ is fixed, using Taylor expansion of $e^{t}$, we can act $D$ term by term is $D$ is linear and the norm of each term is finite, thus $D(e^{t})=e^{t}\neq0$. Can anyone further explain to me?