# Extra Maths: Analysis, Darboux’s Theorem

As before, these questions are aimed at first-year or second-year undergraduates studying a course in Real Analysis.

Question 1. Let $f:[a,b]\longrightarrow\mathbb{R}$ be differentiable, with derivative $f'$. Suppose that $f'(a)>0>f'(b)$. Show there exists $c\in(a,b)$ such that $f'(c)=0$.

Compare Question 1 with Rolle’s Theorem.

Question 2 (Darboux’s Theorem). Let $f:\mathbb{R}\longrightarrow\mathbb{R}$ be differentiable, with derivative $f'$. Show that $f'$ has the intermediate value property.

Compare Question 2 with the Mean Value Theorem.

Question 3. Let $f:[a,b]\longrightarrow\mathbb{R}$ be differentiable, with derivative $f'$. Show that there exists $c\in[a,b]$ such that $f'$ is continuous at $c$.

This is very difficult! See this article. Nevertheless, I would like to expand on this argument in a subsequent post. Watch this space.

Question 4. Suppose that $g:\mathbb{R}\longrightarrow\mathbb{R}$ satisfies the intermediate value property. Need $g$ admit an anti-derivative?

Actually, this one is straightforward, so I’ll give the argument right away.

Proof. Certainly not. In fact we have just seen that every derivative (so, a function that admits an anti-derivative) is continuous somewhere. However, there are plenty of functions that satisfy the intermediate value property which are continuous nowhere. For example, there are functions whose graphs are dense in the plane. $\square$

Remark. For an in-depth discussions of the continuity of derivatives, see this article.