**Taylor Formula and displacement operator:** I (too often) see in papers (mathematical physics but a recent paper by mathematicians also) the statement 

> Let $D=\frac{d}{dx}$ be the derivation operator. Then, for all $f\in C^\infty(\mathbb{R})$, 
$$
e^{tD}[f](x)=f(x+t)
$$

which is false (take any $\phi\in C^\infty(\mathbb{R})$ with compact support, for instance). 

The formula is true for all $t\in \mathbb{R}$ iff $f$ is analytic over $\mathbb{R}$ ($f\in C^\omega(\mathbb{R})$).