Let $f: R \to R$ be a map of class $C^1$. Show the set P = $\left\{x \in R : f'(x) \neq 0\right\}$ is open.
I've been trying to answer this question, but I have no clue. I'd accept any hints.
Let $f: R \to R$ be a map of class $C^1$. Show the set P = $\left\{x \in R : f'(x) \neq 0\right\}$ is open.
The function $f'$ is continuous . So inverse image of closed sets is a closed set. as the singleton $\{0\}$ is closed
$\mathbb{R}\setminus P =(f')^{-1}(\{0\})$ is closed as $f'$ is continuous. So $(\mathbb{R}\setminus (f')^{-1}(\{0\}))=P$ is open.