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

**Question 1.** Let be differentiable, with derivative . Suppose that . Show there exists such that .

Compare Question 1 with Rolle’s Theorem.

**Question 2 (Darboux’s Theorem).** Let be differentiable, with derivative . Show that has the intermediate value property.

Compare Question 2 with the Mean Value Theorem.

**Question 3.** Let be differentiable, with derivative . Show that there exists such that is continuous at .

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 satisfies the intermediate value property. Need 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.

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

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