racodelitusmn

2022-07-07

Let $f:R\to R$ be a map of class ${C}^{1}$. Show the set $P=\left\{x\in R:f\prime \left(x\right)\ne 0\right\}$ is open.

Caiden Barrett

Expert

Step 1
The function f′ is continuous . So inverse image of closed sets is a closed set. as the singleton {0} is closed.
Step 2
$\mathbb{R}\setminus P=\left({f}^{\prime }{\right)}^{-1}\left(\left\{0\right\}\right)$ is closed as f′ is continuous. So $\left(\mathbb{R}\setminus \left({f}^{\prime }{\right)}^{-1}\left(\left\{0\right\}\right)\right)=P$ is open.

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