EDB — 1CP

[1CP] Difficulty:*. Describe a function \(f:ℝ→ℝ\) that is differentiable and such that the image of \([0,1]\) using \(f'\) is \(f'([0,1])=(-1,1)\).

Before looking for the example, ponder on this notions. We remember the Darboux property [1C8]↺↻: the image \(f'(I)\) of an interval \(I\) is an interval; but this does not say that the image of \(f'([0,1])\) should be a closed and bounded interval. If, however, we also knew that \(f'\) is continuous, what could we say of \(f'([0,1])\)? So what do you deduce a priori about the sought example?