Say $f: \mathbb R \to \mathbb R$ is nowhere zero (like e.g. the constant map 1). Is the map $x \mapsto x f(x)$ a diffeomorphism?
It seems to me that the answer is no because the derivative of a diffeomorphism on $\mathbb R$ is non zero everywhere.
But on the other hand I don't understand why for example $x \mapsto 2 x$ is not a diffeomorphism!
Please could someone resolve my confusion?
No, consider $f(x)=\frac1x$.
To address the last point, $x \mapsto 2 x$ is a diffeomorphism.