Find two discontinuous functions whose product is continuous

Question

Find $f: \mathbb{R}\rightarrow \mathbb{R}$ and $g: \mathbb{R}\rightarrow \mathbb{R}$ such that $f$ is not continuous at 0, $g$ is not continuous at 3, and $f(x)g(x)$ is continuous everywhere.

1. What does $f: \mathbb{R}\rightarrow \mathbb{R}$ mean? Does it necessarily mean that the domain of the function is $\mathbb{R}$?
2. Where do I start to solve this problem? I was thinking $f(x)= \frac{x-3}{x}$ and $g(x)=\frac{x}{x-3}$ but is it correct if $f: \mathbb{R}\rightarrow \mathbb{R}$ and $g: \mathbb{R}\rightarrow \mathbb{R}$?

Answer 1

Hint: Find functions $f(x)$ and $g(x)$ such that their product $f(x)g(x)$ equals the constant function $0$ but where $f(x)$ is discontinuous at $x = 0$ and $g(x)$ is discontinuous at $x = 3$.