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 $t$ iff $f$ is entire.