EDB — 1C8

Exercises

1. [1C8] Prerequisites:[1C6]↺↻.Note:Darboux properties.

   Let \(A⊆ ℝ\) be an open set, and suppose that \(f:A→ℝ\) is differentiable. We want to show that, for each interval \(I⊂ A\), the image \(f'(I)\) is an interval.

   So prove this result. For \(x,y∈ I\) with \(x{\lt}y\), let’s define \(a=f'(x), b= f'(y)\). Let’s assume for simplicity that \(a{\lt}b\). For any \(c\) with \(a{\lt} c {\lt} b\), there exists \(𝜉∈ I\) with \(x{\lt}𝜉{\lt}y\) such that \(f'(𝜉)=c\).

   (Finally, show that this property actually implies that the image \(f'(I)\) of an interval \(I\) is an interval.)

   Solution 1

   [1C9]↺↻