Okay so I'm new to calculus and still trying to grasp what I can and cannot do, and was wondering if the way I showed my counter example is correct, notably the domain part.. Also I'm desperate for tips on how to prove/disprove these type of questions and I would be extremely thankful if you could show me different patterns of proofs as I want to absorb as much as I can!
You did a great job :)
Writing $x \in {\Bbb R}\setminus \{x_0\}\{0\}$ is not defined. You should have just written $x \in {\Bbb R}\setminus \{x_0\}$, where $x_0 = 0$.