Is $f'-f$ darboux?

Question

Suppose $f:[0,1]\to\mathbb{R}$ is continuous and (not necessarily continuously) differentiable on $(0,1)$. Darboux's theorem tells us that the intermediate value theorem holds for $f'$. Does the intermediate value theorem hold for $f'-f$?

More generally, does $f$ continuous, $g$ darboux imply $f+g$ darboux?

Answer 1

For the first part apply Darboux's theorem on $$h(x) = f(x) - \int_{0}^x f(t)\mathrm dt$$