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