If I have a path $f:[0,1]\to S^1$. How do I show that the image of the set of critical points is closed and has an empty interior? The hint suggests covering the set of critical points with a finite number of disjoint open intervals where the derivative of $f$ is within $\epsilon$-balls in $[0,1]$ and use its images to cover the image. However, I don't know how to apply it. Can someone help?
Edit: $f$ is a $C^1$ function so I get why the set is closed. The trouble I have is with proving it has an empty interior.