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.

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