Homomorphism Notation

Question

I have a question on notation.

The question in my text: A function $f:\mathbb{R}\rightarrow\mathbb{R}^{\times}$ is a homomorphism if and only if $f(x+y)=f(x)+f(y)$ for all $x,y\in \mathbb{R}$.

My question is what does $\mathbb{R}^{\times}$ mean?

Answer 1

$\mathbb R^\times$ is the set of invertible real numbers, that is, $$\mathbb R^\times = \{x\in\mathbb R: x^{-1}\text{ exists}\} = \{x\in\mathbb R: x\ne 0\}$$ However, $(\mathbb R^\times,+)$ is not a group, therefore I strongly suspect that the equation in your text actually reads $f(x+y) = f(x)f(y)$, with multiplication instead of addition on the right hand side.